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The Euler zeta function is defined by ζ E (s)=P ∞n=1(−1) n−1s .The purpose of this paper is to find formulas of the Euler zeta func-tion’s values. In this paper, for s ∈ N we find the recurrence formula ofζ E (2s) using the Fourier series. Also we find the recurrence formula ofP ∞n=1(−1) n−1 (2n−1) 2s−1 , where s ≥ 2(∈ N). 1. IntroductionThe Euler zeta function is defined by ζ E (s) =P ∞n=1(−1) n n s (see [3, 4]). Inthis paper we investigate the recurrence formula of the Euler zeta function fors = 2n with Fourier series. By this result we can find ζ E (2n) for all n ∈ N.For s ∈ C, the Riemann zeta function or the Euler-Riemann zeta function,ζ(s) is defined byζ(s) =X ∞n=1 1n s (s ∈ C), (see [5, 6])which converges when the real part of s is greater than 1. R. Ap´ery provedthat the number ζ(3) is irrational. But it is still an open problem to proveirrationality of ζ(2k +1) for the long time.As well known special values, for any positive even number 2n,ζ(2n) = (−1) n+1 B 2n (2π) 2n 2(2n)!, (see [1])where B

In this paper, we give a necessary and sufficient condition for the hyponormality of the block Toeplitz operators T�, where � = F +G∗, F(z), G(z) are some matrix valued polynomials on the vector valued Bergman space L2(D,Cn). We also show some necessary conditions for the hyponormality of TF+Gwith F + G∗ 2 h∞ Mn×n on L2(D,Cn). Let D and T be the open unit disk and unit circle in the complex plane C re- spectively and dA be the normalized Lebesgue area measure on D. h ∞ denotes the space of all bounded harmonic functions on D. L ∞ (D,dA) and L 2 (D,dA) denote the space of essential bounded measurable functions and the space of the square integrable functions on D with respect to dA, respectively. The Bergman space L 2 consists of all analytic functions in L 2 (D,dA). We denote the space of vector valued square integrable functions on D by L 2 (D,C n ) = L 2 (D,dA)⊗C n and the vector valued Bergman space on D by L 2(D,C n ) = L 2 ⊗ C n , respec- tively, where ⊗ denotes the Hilbert space tensor product. In this paper F T denotes the transpose of the matrix F and G ∗ denotes the adjoint of the matrix G.

In this paper, we introduce the q-analogue of p-adic log gamma functions with weight alpha. Moreover, we give a relationship be- tween weighted p-adic q-log gamma functions and q-extension of Genocchi and Euler numbers with weight alpha. Assume that p is a fixed odd prime number. Throughout this paper Z, Zp, Qp and Cp will denote the ring of integers, the field of p-adic rational numbers and the completion of the algebraic closure of Qp, respectively. Also we denote N � = N ( {0} and exp(x) = e x . Let vp : Cp ! Q ( {1} (Q is the field of rational numbers) denote the p-adic valuation of Cp normalized so that vp (p) = 1. The absolute value on Cp will be denoted as |·|, and |x| p = p v p(x)

Even though there is a formula for h 0 ( X,O X ( nK X )) for acanonical threefold X , it is not easy to compute h 0 ( X,O X ( nK X )) be-cause the formula has a term due to singularities. In this paper, we finda way to control the term due to singularities. We show nonvanishing ofplurigenus for the case when the index r in the singularity type 1 r (1 ,− 1 ,b )is sufficiently large. Throughout this paper X is assumed to be a projective threefold with onlycanonical singularities and an ample canonical divisor K X over the complexnumber field C, i.e., a canonical threefold.It is well known that H 0 ( X,O X ( mK X )) does not vanish and generates abirational map for a sufficiently large m . When X is a surface of general type, H 0 ( X,O X ( mK X )) does not vanish for m ≥ 2 and H 0 ( X,O X ( mK X )) generatesa birational map for m ≥ 5.In a case of a threefold of general type, M. Reid and A. R. Fletcher de-scribed the formula for χ ( O X ( nK X )) (see [1]). Combining the formula for χ

For convex bodies, chord power integrals were introduced and studied in several papers (see [3], [6], [14], [15], etc.). The aim of this article is to study them further, that is, we establish the BrunnMinkowski-type inequalities and get the upper bound for chord power integrals of convex bodies. Finally, we get the famous Zhang projection inequality as a corollary. Here, it is deserved to mention that we make use of a completely distinct method, that is using the theory of inclusion measure, to establish the inequality. 1. Preliminaries The setting for this paper is n-dimensional Euclidean space R. We will denote by convex figure a compact convex subset of R, and by convex body a convex figure with nonempty interior. Let Sn−1 denote the unit sphere centered at the origin o in R, and write αn−1 for the (n − 1)-dimensional volume of Sn−1. Let Bn be the closed unit ball in R, write ωn for the n-dimensional volume of Bn. Note that ωn = 2π n 2 nΓ(n2 ) , αn−1 = nωn. By a direction, we mean a unit vector, that is, an element of Sn−1. If u is a direction, we denote by u⊥ the (n − 1)-dimensional subspace orthogonal to u and by lu the line through the origin parallel to u. Denote by AGi,n the affine Grassmann manifold of i-dimensional planes in R. It is a homogeneous space under the action of the motion group G(n) (see [7], p.199). Let dξk be the normalized invariant measure of AGi,n whose restriction to the Grassmann manifold Gi,n is the invariant probability measure. Let ξ1 be a random line intersecting K. Then vol1(K ∩ ξ1) is the chord length of the intersection K ∩ ξ1. The chord power integrals of K are defined by Iλ(K) = 2αn−1 n ∫ ξ1∈AG1,n vol1(K ∩ ξ1)dξ1, 0 ≤ λ < ∞. Received November 30, 2006; Revised April 5, 2007. 2000 Mathematics Subject Classification. 52A40.

In this paper, we introduce the SP-integral and the SPfi-integral deflned on an interval in the n-dimensional Euclidean space R n . We also investigate the relationship between these two integrals. It is well known (3) that the Perron integral deflned on an interval of the real line R by major and minor functions which are not assumed to be continuous is equivalent to the one deflned by continuous major and minor functions and that the strong Perron integral deflned on an interval of R by strong major and minor functions is equivalent to the McShane integral. In this paper, we introduce Perron-type integrals deflned on an inter- val of the n-dimensional Euclidean spaceR n using the strong major and minor functions, and investigate the relationship between these integrals. We shall call it the strong Perron integral, or brie∞y SP-integral. For a subset E of the n-dimensional Euclidean spaceR n , the Lebesgue measure of E is denoted by jEj. For a point x = (x1;x2;¢¢¢ ;xn) 2R n , the norm of x is kxk = max1•in jxij and the --neighborhood U(x;-) of x is an open cube centered at x with sides equal to 2-. For an interval I = (a1;b1)£(a2;b2)£¢¢¢(an;bn) ofR n with ai fi(fi 2 (0;1)), then the interval I is said to be fi-regular.

In this paper we begin the study of Lk-2-type hypersurfaces of a hypersphere S n+1 � R n+2 for k � 1. Let : M 3 ! S 4 be an ori- entable Hk-hypersurface, which is not an open portion of a hypersphere. ThenM3 is ofLk-2-type if and only ifM3 is a Clifford tori S1(r1)×S2(r2), r2 +r 2 = 1, for appropriate radii, or a tube T r(V 2) of appropriate con- stant radiusr around the Veronese embedding of the real projective plane RP 2( p 3).

Let x : M n−1 → R n (n ≥ 4) be an umbilical free hyper-surface with non-zero principal curvatures. Then x is associated witha Laguerre metric g, a Laguerre tensor L, a Laguerre form C, and aLaguerre second fundamental form B, which are invariants of x underLaguerre transformation group. We denote the Laguerre scalar curvatureby R and the trace-free Laguerre tensor by L˜ := L− 1n−1 tr(L)g. Inthis paper, we prove a local classification result under the assumption ofparallel Laguerre form and an inequality of the typekL˜k ≤ cR,where c= 1(n−3) √ (n−2)(n−1) is appropriate real constant, depending onthe dimension. 1. IntroductionLet x : M n−1 → R n be an umbilical free hypersurface with non-zero princi-pal curvatures. Let ξ : M → S n−1 be its unit normal. Let {e 1 ,e 2 ,...,e n−1 }be the orthonormal basis for TM with respect to dx · dx, consisting of unitprincipal vectors. Let r i = 1k i , r = r 1 +r 2 +···+r n−1 n−1 be the curvature radius andmean curvature radius of x respectively, where k

In this paper, we consider the problem of existence of confor- mal metrics with prescribed mean curvature on the unit ball of Rn, n � 3. Under the assumption that the order of flatness at critical points of pre- scribed mean curvature function H(x) is � 2)1,n 2), we give precise estimates on the losses of the compactness and we prove new existence result through an Euler-Hopf type formula. 1. Introduction and main result In this article, we consider the problem of existence of conformal scalar flat metric with prescribed boundary mean curvature on the standard n-dim- ensional ball. Let B n be the unit ball in R n , n ≥ 3, with Euclidean metric g0. Its boundary will be denoted by S n−1 and will be endowed with the standard metric still denoted by g0. Let H : S n−1 → R be a given function, we study the problem of finding a conformal metric g = u 4 n−2g 0 such that Rg = 0 in B n and hg = H on S n−1 . Here Rg is the scalar curvature of the metric g in B n and hg is the mean curvature of g on S n−1 . This problem is equivalent to solving the following nonlinear boundary value equation: (P) ( �u = 0 in B n ∂u ∂ν + n − 2 2 u = n − 2 2 Hu n n−2 on S n−1

In this paper, we investigate p-biharmonic maps u : (M,g) →(N,h) from a Riemannian manifold into a Riemannian manifold with non-positive sectional curvature. We obtain that ifR M |τ(u)| a+p dv g < ∞ andR M |d(u)| 2 dv g < ∞, then u is harmonic, where a ≥ 0 is a nonnegativeconstant and p ≥ 2. We also obtain that any weakly convex p-biharmonichypersurfaces in space formN(c) with c ≤ 0 is minimal. These results giveaffirmative partial answer to Conjecture 2 (generalized Chen’s conjecturefor p-biharmonic submanifolds). 1. IntroductionHarmonic maps play a central roll in geometry. They are critical pointsof the energy E(u) =R M|du| 2 2 dv g for smooth maps between manifolds u :(M,g) → (N,h) and the Euler-Lagrange equation is that tension field τ(u)vanishes. Extensions to the notions of p-harmonic maps, F-harmonic mapsand f-harmonic maps were introduced and many results have been carried out(for instance, see [1, 2, 3, 8, 23]). In 1983, J. Eells and L. Lemaire [10] proposedthe problem to consider the biharmonic maps: they are critical maps of thefunctionalE

Let λ f (n) denote the n-th normalized Fourier coefficient ofa primitive holomorphic form f for the full modular group Γ = SL 2 (Z).In this paper, we are concerned with Ω-result on the summatorPy function n6x λ 2f (n 2 ), and establish the following resultX n6x λ 2f (n 2 ) = c 1 x + Ω(x 49 ),where c 1 is a suitable constant. 1. Introduction and main resultsAccording to the Langlands program, there are many hidden structures un-derlying the Fourier coefficients of an automorphic form. Thus it is very impor-tant and essential to investigateits summatoryfunction overa certain sequence.Let H ∗ k be the set of all normalized Hecke eigencuspforms of even integralweight kfor the full modular group Γ = SL 2 (Z). For f(z) ∈ H ∗ k , f(z) has thefollowing Fourier expansion at the cusp ∞f(z) =X ∞n=1 λ f (n)n k−12 e 2πinz ,where λ f (n) is real and satisfies the multiplicative property(1.1) λ f (m)λ f (n) =X d|(m,n) λ f mnd 2 for any integers m≥ 1 and n≥ 1.The size and oscillations of λ f (n) deserve deep research. In 1974, Deligne

In this paper, we investigate relationships among property(k-β), weak property (β k ), k-nearly uniformly convexity and property(A k ). 1. IntroductionLet (X,k·k) be a real Banach space and X ∗ the dual space of X. By B X andS X , we denote the closed unit ball of X and the unit sphere of X, respectively.For a sequence (x n ) in X, we let sep(x n ) = inf {kx n −x m k : n 6= m}. Denoteby N and R the set of natural numbers and real numbers, respectively. ForA, B ⊂ N, we write A 0, thereexists δ > 0 such that if x,y ∈ B

In 2005 Kadiri proved that the Riemann zeta function ζ(s)does not vanish in the regionRe(s) ≥ 1 −1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 5.69693. In this paper we will show that R 0 can be taken R 0 =5.68371 using Kadiri’smethod together with Platt’s numerical verificationof Riemann Hypothesis. 1. IntroductionIt is well known that the distribution of prime numbers is deeply relatedwith the zeros of the Riemann zeta function ζ(s). Let π(x) be the number ofprimes up to x. Hadamard and de la Vall´ee Poussin proved the prime numbertheorem which states that π(x) is asymptotic to Li(x) =R x2dtlogt . In 1899, dela Vall´ee Poussin proved that ζ(s) does not vanish in the regionRe(s) ≥ 1−1R 0 log|Im(s)|, |Im(s)| ≥ 2with R 0 = 34.82. From this zero-free region for ζ(s) the error term π(x)−Li(x)can be estimated:π(x) −Li(x) = O x exp −rlogxR 0 when x → ∞.See [18] and Section 1 of [7]. The constant R 0 has been improved by manyauthors, particularly by Rosser [13, 14, 15], Schoenfeld [14, 15], Stechkin [16],Ford [1, 2], and Kadiri [7]. Recently Kadiri improved significantly on R

In this paper, we study some quantity equivalent to the normof Bloch to A pα composition operator where A pα is the weighted Bergmanspace on the unit ball of C n (0 < p < ∞ and −1 < α < ∞). 1. IntroductionLet n ≥ 1 be an integer. Let H(B) denote the space of all holomorphicfunctions in the unit ball B ≡ B n of the complex n-dimensional Euclideanspace C n . Let U stand for B 1 . The Bloch space B = B(U) is defined byB = {f ∈ H(U) : sup z∈U |f ′ (z)|(1 − |z| 2 ) < ∞}. With the norm kfk B =|f(0)|+sup z∈U |f ′ (z)|(1−|z| 2 ), B is a Banach space. Let ν denote the Lebesguemeasure on C n , so normalized that ν(B) = 1. For −1 < α < ∞, we setc α = Γ(n+α+1)/{Γ(n+1)Γ(α+1)} and dν α (z) = c α (1−|z| 2 ) α dν(z), z ∈ B.Note that ν α (B) = 1. For 0 < p < ∞ and −1 < α < ∞, the weighted Bergmanspace A pα ≡ A pα (B) is defined by A pα = {f ∈ H(B) :R B |f(z)| dν (z) < ∞}.For f ∈ A pα , we write kfk A pα =R B |f(z)| p dν α (z) 1p .In 2003, Kwon and Lee proved the following theorem. This result relates tothe pull-back property (see [1], [2], [5], [7] for example).Theorem 1 ([6]). Let f : U → U be a holomorphic function with f(0) = 0.For 1 ≤ p < ∞ and −1 < α < ∞, the bounded operator C

In this paper, a boundary version of the Schwarz and Carath-´eodory inequality are investigated. New inequalities of the Carath´eodory’sinequality and Schwarz lemma at boundary are obtained by taking intoaccount zeros of f(z) function which are different from zero. The sharp-ness of these inequalities is also proved. 1. IntroductionLet f be a function which is holomorphic on the D : {z : |z| < 1} and vanishat z = 0, and suppose that |f| < 1 for all z ∈ D. Then the inequality(1.1) |f(z)| ≤ |z|holds for all z ∈ D, and moreover(1.2) |f ′ (0)| ≤ 1.Equality is achieved in (1.1) (for some nonzero z ∈ D) or in (1.2) if and onlyif f is an entire linear function of the form f(z) = e iα z, where α is a realnumber([2], p. 381).Let the zeros of f be z 1 ,z 2 ,...,z n . If we apply inequality (1.1) to the func-tion f(z)Q nk=1 h 1−z k zz+z k i, we can conclude in the following Schwarz’s inequality:(1.3) |f(z)| ≤ |z|Y nk=1 z −z k 1−z k z and|f ′ (0)| ≤Y nk=1 |z k |. Received May 3, 2013.2010 Mathematics Subject Classification. Primary 30C80.Key words and phrases. Schwarz lemma on the boundary, holomorphic function, Julia-Wolff-Lemma.

In this paper we investigate (n+1)(n ≥ 3)-dimensional con-tact CR-submanifolds M of (n−1) contact CR-dimension in a completesimply connected Sasakian space form of constant φ-holomorphic sec-tional curvature c 6= −3 which satisfy the condition h(FX,Y )+h(X,FY )= 0 for any vector fields X,Y tangent to M, where h and F denote thesecond fundamental form and a skew-symmetric endomorphism (definedby (2.3)) acting on tangent space of M, respectively. 1. IntroductionLet S 2m+1 be a (2m + 1)-unit sphere in the complex (m + 1)-space C m+1 ,i.e.,S 2m+1 := {(z 1 ,...,z m+1 ) ∈ C m+1 | m X +1j=1 |z j | 2 = 1}.For any point z ∈ S 2m+1 we put ξ = Jz, where J denotes the complex structureof C m+1 . Denoting by π the orthogonal projection : T z C m+1 → T z S 2m+1 andputting φ = π◦J, we can see that the set (φ,ξ,η,g) defines a Sasakian structureon S 2m+1 , where g is the standard metric on S 2m+1 induced from that of C m+1 and η is a 1-form dual to ξ. Hence S 2m+1 can be considered as a Sasakianmanifold of constant curvature 1 (cf. [2, 4, 5, 6, 7, 8, 9]).Let M be an (n+1)-dimensional submanifold tangent to the structure vectorfield ξ of S

In this paper we consider the existence of smooth conics ona general hypersurface of degree din P n . 1. IntroductionLet X be a generalhypersurfaceof degreed in a complex projective space P n .Recall that R e (X) is the open subscheme of the Hilbert scheme Hilb et+1 (X)which parameterizes smooth rational curves of degree e lying on X. Recently J.Harris, M. Roth, and J. Starr obtained the following general result for d 2 and d 2n − 3, then R 1 (X) is empty. Ifd ≤ 2n−3,thenR 1 (X) issmoothofdimension2n−3−d.In this paperwe will investigatethe nonemptynessandsmoothness ofR 2 (X).Theorem 1.1. LetX beageneralhypersurfaceofdegreed inP

This paper generalizes the Aleksandrov problem and MazurUlam theorem to the case of n-normed spaces. For real n-normed spacesX and Y, we will prove that f is an affine isometry when the mappingsatisfies the weaker assumptions that preserves unit distance, n-colinearand 2-colinear on same-order. 1. IntroductionLet X and Y be metric spaces. A mapping f : X → Y is called an isometryif f satisfies d Y (f(x),f(y)) = d X (x,y) for all x,y ∈ X, where d X (·,·) andd Y (·,·) denote the metrics in the spaces X and Y , respectively. For some fixednumber r > 0, assume that f preserves the distance r, i.e., for all x,y ∈ X withd X (x,y) = r, it holds that d Y (f(x),f(y)) = r. Then r is called a conservative(or preserved) distance for the mapping f.In 1970, Aleksandrov [1] posed the following problem: Examine whether theexistence of a single conservative distance for some mapping T implies that Tis an isometry.In 1932, Mazur and Ulam [8] proved a theorem that every isometry of a realnormed space onto a real normed space is a linear mapping up to a translation.In 1993, Rassias and Semrl [10] proved a series of results on the DOPPˇ(distance one preserving property) for the normed spaces (see also [6, 7, 9]).Since 2004, the Aleksandrovproblem has been discussedin the n-normed spaces(n ≥ 2) (see [2, 3, 4, 5]). Chu, Lee and Park proved:Theorem 1.1 ([3]). Let X and Y be two real linear n-normed spaces. If amapping f : X → Y satisfies the following conditions:(1) f has the n-DOPP;(2) f is n-Lipschitz

In this paper, a finite difference scheme for a two-point boun-dary value problem of 2nth-order ordinary differential equations is pre-sented. The convergence and uniqueness of the solution for the schemeare proved by means of theories on matrix eigenvalues and norm. Nu-merical examples show that our method is very simple and effective, andthat this method can be used effectively for other types of boundary valueproblems. 1. IntroductionAssume that a 2nth-order (n ≥ 2) two-point boundary valve problem (BVP)is given in the formy (2n) = f(x)y +g(x), 0 < x < 1,y (2α) (0) = a α , α = 0,1,...,n−1,y (2α) (1) = b α ,(1.1)where f(x) and g(x) are continuous functions on the closed interval [0,1] anda α ,b α are all real constants. The problem (1.1) frequently occurs in structuralengineering, astrophysics and other branches of physical sciences. The problem(1.1) has a unique analytic solution for a fixed positive integer n ≥ 2 under thegeneral condition [5](1.2) f(x) 6= ( −1) n j 2n

In this paper, we give sufficient conditions on a canonical genus 4 curve for it to be Chow (semi)stable. A Deligne-Mumford stable curve is a complete connected curve C having ample dualising sheaf !C and admitting only nodes as singularities. An n- canonical curve C ⊂ P N is a Deligne-Mumford stable curve of arithmetic genus g embedded by the complete linear system |! ⊗n C | where N = (2n−1)(g−1)−1 if n ≥ 2, and N = g − 1 if n = 1. Let Chowg,n be the closure of the locus of the Chow forms of n-canonical curves of arithmetic genus g in the Chow variety of algebraic cycles of dimension 1 and degree 2g−2 in P N. The natural action of SLN+1 on P N induces an action on Chowg,n. Denote the corresponding GIT (Geometric Invariant Theory) quotient space by Chowg,n//SLN+1. To understand this quotient space as a parameter space of curves with some geometric properties, we need to find Chow stability conditions. Mumford showed that, for n ≥ 5 and g ≥ 2, the Chow stable curves are pre- cisely Deligne-Mumford stable curves and there is no strictly Chow semistable curve (cf. (14)). This implies that the quotient space is precisely the moduli space of Deligne-Mumford stable curves M4. The cases when n = 3 and g ≥ 3 were concerned by Schubert in (16). He proved that a 3-canonical curve of genus g ≥ 3 is Chow stable if and only if it is pseudo-stable and also showed that there is no strictly Chow semistable curve, and thus the quotient space is the moduli space of pseudo-stable curves M ps . A pseudo-stable curve is a complete connected curve C satisfying the following properties.

Approximate fixed point property problem for Mann itera-tion sequence of a nonexpansive mapping has been resolved on a Banachspace independent of uniform (strict) convexity by Ishikawa [Fixed pointsand iteration of a nonexpansive mapping in a Banach space, Proc. Amer.Math. Soc. 59 (1976), 65–71]. In this paper, we solve this problem fora class of mappings wider than the class of asymptotically nonexpan-sive mappings on an arbitrary normed space. Our results generalize andextend several known results. 1. IntroductionLet C be a nonempty subset of a normed space X and T : C → C be amapping. Then T is said to be(1) Lipschitzian if for each n ∈ N, there exists a positive real number k n such thatkT n x−T n yk ≤ k n kx−ykfor all x,y ∈ C,(2) uniformly k-Lipschitzian if T is Lipschitzian with sequence {k n } suchthat k n = k for all n ∈ N,(3) asymptotically nonexpansive ([8]) if T is Lipschitzian with sequence {k n }such that k n ≥ 1 for all n ∈ Nwith lim n→∞ k n = 1.Clearly, every nonexpansive mapping T (i.e., kTx − Tyk ≤ kx − yk forall x,y ∈ C) is asymptotically nonexpansive with sequence {1} and everyasymptotically nonexpansive mapping is uniformly k-Lipschitzian with k =sup

Let (M 3 ,g) be a 3-dimensional closed Sasakian spin mani- fold. Let Smin denote the minimum of the scalar curvature of (M 3 ,g). Let � + > 0 be the first positive eigenvalue of the Dirac operator of (M 3 ,g). We proved in (13) that if � + belongs to the interval � + 2 1 , 5 � , then � + satisfies � + � Smin+6 8 . In this paper, we remove the restriction "if � + belongs to the interval � + 2 ( 1 , 5 )" and prove

In this paper, the Schur convexity is generalized to Schurf-convexity, which contains the Schur geometrical convexity, harmonicconvexity and so on. When f : R + → Ris defined as f(x) = (x m −1)/mif m6= 0 and f(x) = lnxif m= 0, the necessary and sufficient conditionsfor f-convexity (is called Schur m-power convexity) of Gini means aregiven, which generalize and unify certain known results. 1. IntroductionLet p,q ∈ Rand a,b ∈ R + := (0,∞). The Gini means [13] are defined as(1.1) G p,q (a,b) = a p +b p a q +b q 1/(p−q) , p 6= q,exp a p lna +b p lnba p +b p , p = q.It is easy to see that the Gini means G p,q (a,b) are continuous on the domain{(a,b;p,q) : a,b ∈ R + ;p,q ∈ R} and differentiable with respect to (a,b) ∈ R 2+ for fixed p,q ∈ R. Also, Gini means are symmetric with respect to a,b and p,q.Gini means G p,q (a,b) contain many classical two variable means, for ex-ample, G 1,0 = A is the arithmetic mean, G 0,0 = G is the geometric mean,G −1,0 = H is the harmonic mean, and more generally, the p-th power meanis equal to G

In this note, the q-extension of the twisted Lerch Euler zetafunctions considered by Jang [Bull. Korean Math. Soc. 47 (2010), no.6, 1181–1188] is further investigated, and the generalized multiplicationtheorem for the q-extension of the twisted Lerch Euler zeta functionsis given. As applications, some well-known results in the references arededuced as special cases. 1. IntroductionThroughout this paper, Z p , Q p and C p will denote the ring of p-adic rationalintegers, the field of p-adic rational numbers and the completion of the algebraicclosure of Q p , respectively. Let v p be the normalized exponential valuation ofC p with |p| p = p −v p (p) = p −1 . When one talks of q-extension, q is variouslyconsidered as an indeterminate, a complex number q ∈ Cor a p-adic numberq ∈ C p . If q ∈ C, one normally assume |q| < 1. If q ∈ C p , then we normallyassume |q − 1| p < p 11−p , so that q x = exp(xlogq) for each x ∈ Z p . Forf ∈ UD(Z p ,C p ) = {f | f : Z p → C p is the uniformly differentiable function},the p-adic q-integral (also be called as q-Volkenborn integration) is defined by(see [6, 13])(1.1) I

In this paper, we first talk about a more wide class of non-linear mappings, Then, we deal with weak convergence theorems for gen-eralized hybrid mappings in a Hilbert space. 1. IntroductionLet C be a nonempty closed convex subset of a real Hilbert space H. Thena mapping T :C → C is said to be nonexpansive if kTx − Tyk ≤ kx − ykfor all x,y ∈ C. The set of fixed points of T is denoted by F(T). A mappingT : C → C with F(T) 6= ∅ is called quasi-nonexpansive if kx −Tyk ≤ kx −ykfor all x ∈ F(T) and y ∈ C. It is well-known that the set F(T) of fixed pointsof a quasi-nonexpansive mapping T is closed and convex; see [10].A important example of nonexpansivemappings in a Hilbert space is a firmlynonexpansive mapping ifkFx −Fyk 2 ≤ hx −y,Fx−Fyifor all x,y ∈ C; see for instance, [3, 5]. It is known that a mapping F : C → Cis firmly nonexpansive if and only ifkFx−Fyk 2 +k(I −F)x −(I −F)yk 2 ≤ kx−yk 2 for all x,y ∈ C, where I is the identity mapping on H. It is also known that afirmly nonexpansive mapping F can be deduced from an equilibrium problemin a Hilbert space; see, for instance, [2, 4].Recently, Kohsaka and Takahashi [11] introduced the following nonlinearmapping: Let E be a smooth, strictly convex and reflexive Banach space, letJ be the duality mapping of E and let C be a nonempty closed convex subsetof E. Then, a mapping S : C → C is said to be nonspreading ifφ(Sx,Sy)+φ(Sy,Sx) ≤ φ(Sx,y)+φ(Sy,x)

In this paper, the reverse Bonnesen style inequalities for con- vex domain in the Euclidean plane R 2 are investigated. The Minkowski mixed convex set of two convex sets K and L is studied and some new geo- metric inequalities are obtained. From these inequalities obtained, some isoperimetric deficit upper limits, that is, the reverse Bonnesen style in- equalities for convex domain K are obtained. These isoperimetric deficit upper limits obtained are more fundamental than the known results of Bottema ((5)) and Pleijel ((22)).

In the present paper, we discuss the rigidity phenomenon ofclosed minimal submanifolds in a locally symmetric Riemannian manifoldwith pinched sectional curvature. We show that if the sectional curvatureof the submanifold is no less than an explicitly given constant, then eitherthe submanifold is totally geodesic, or the ambient space is a sphere andthe submanifold is isometric to a product of two spheres or the Veronesesurface in S 4 . 1. IntroductionLet M n be an n-dimensional closed minimal submanifold in an (n + p)-dimensional Riemannian manifold N n+p . We denote by S the squared normof the second fundamental form of M. If N n+p is the (n + p)-dimensionalunit sphere S n+p , a famous rigidity theorem due to Simons [11], Lawson [8]and Chern-do Carmo-Kobayashi [1] says that if S ≤ n2−1/p , then either M istotally geodesic, or M is one of the Clifford minimal hypersurfaces S k q kn ×S n−k q n−kn , k = 1,...,n − 1, or n = 2, p = 2, and M is the Veronesesurface in S 4 . Further discussions have been carried out by many other authors[2, 9, 12, 13, 14, 15], etc.Denote by K

We prove a Hardy inequality related to Carnot-Carath´eodorydistance on H-type groups based on a representation formula on suchgroups. 1. IntroductionThe Hardy inequality in R N states that, for all u ∈ C ∞0 (R N ) and N ≥ 3,(1.1)Z R N |∇u| 2 dx ≥(N −2) 2 4Z R N u 2 |x| 2 dx.Inequality (1.1) has been generalized to Heisenberg group H n and other nilpo-tent groups by several authors ([5, 6, 7, 8, 10, 11, 21]). Especially, the followingHardy inequality holds for 1 < p < 2n and u ∈ C ∞0 (H n ) (see [5], Theorem 3.3),(1.2)Z H n |∇ H u| p ≥ 2n−pp p Z H n |u| p d p ,where d is the Koranyi-Folland non-isotropic gauge. We note there is anotherdistance, called Carnot-Carath´eodory distance on H n and this distance playsan important role in the study of hypoelliptic heat kernel (see [1, 9, 12, 13, 15,16, 17]). Since d cc (ξ) ≥ d(ξ) for all ξ ∈ H n (see [18], Lemma 5.1), inequality(1.2) imply the following(1.3)Z H n |∇ H u| p ≥ 2n−pp p Z H n |u| p d pcc .However, the constant in (1.3) is not sharp.The aim of this paper is to give a new proof of Hardy inequalities associatedwith d

We prove some sharp extremal distance results for functionsin weighted Bergman spaces on the upper halfplane. We also prove newanalogous results in the context of bounded strictly pseudoconvex do-mains with smooth boundary. 1. IntroductionIf Y is a normed space and X ⊂ Y , then we set dist Y (f,X) = inf g∈X kf −gk Y . If the space Y is clear from the context, we write simply dist(f,X). Inthe problems we are going to consider, X itself is going to be a (quasi)-Banachspace.We denote by H(Ω) the space of all holomorphic functions on an open setΩ ⊂ C n . In this paper we consider distance problems in weighted Bergmanspaces over the upper half-plane C + = {x + iy : y > 0} and over a boundedstrictly pseudoconvex domain Ω ⊂ C n with smooth boundary.The weighted Bergman space A pα (C + ) consists of all functions f ∈ H(C )such thatkfk p,α = Z ∞0 Z ∞−∞ |f(x +iy)| p y α dxdy 1p −1 and 0 < p < ∞ (see [5] and [6]). The above spaces are Banachspaces for p ≥ 1 and complete metric spaces for 0 < p < 1. It is naturalto consider the space A

A new Hardy-Hilberttype integral inequality for double se-rieswithweights canbeestablishedbyintroducingaparameter λ (withλ > 1− 2pq )andaweight function ofthe formx 1− 2r (with r > 1). Andthe constant factors of new inequalities established areproved tobe thebestpossible. Inparticular,forcaser =2,anewHilberttypeinequalityisobtained. Asapplications,anequivalentformisconsidered. 1. IntroductionLet {a n } and {b n } be non-negative sequences of real numbers, 1p + 1q = 1and p > 1. IfP ∞n=1 a pn < + ∞ andP ∞n=1 b q < + ∞, then(1.1)X ∞m=1 X ∞n=1 ln mn a m b n m−n≤ πsin πp ! 2 X ∞n=1 a pn ! 1p X ∞n=1 b qn ! q and(1.2)X ∞m=1 X ∞n=1 a m b n m+n≤πsin πp X ∞n=1 a pn ! 1p X ∞n=1 b qn ! 1q ,where the constant factors ( πsin πp ) 2 in (1.1) and πsin p in (1.2) are the bestpossible. And the equalities in (1.1) and (1.2) hold if and only if {a n }, or {b n }is a zero-sequence. They are the famous Hardy-Hilbert inequalities (see [6]),Owing to the importance of the Hardy-Hilbert inequality in analysis andapplications, some mathematicians have been studying them. In particular,some excellent results of (1.2) appear in a lot of the articles (such as [1, 2, 3, 4, 8]etc.). However, the research articles of (1.1) are few. The purpose of thepresent paper is to establish an extension of (1.1), and to prove the constantfactor to be the best possible. And then some important and especial resultsare enumerated, and an equivalent form is considered.

It is well known that one could use a xed loxodromic orparabolic element of a non-elementary group G ˆ M (R n ) as a test mapto test the discreteness of G . In this paper, we show that a test mapneed not be in G . We also construct an example to show that the similarresult using an elliptic element as a test map does not hold. 1. IntroductionThe discreteness has been a fundamental and important problem in M (R n )since J˝rgensen [2] established the following well-known discreteness criterionin 1976.Theorem J ([2, Proposition 2]) : A non-elementary subgroup G of M (R 2 ) isdiscrete if and only if each two-generator subgroup of G is discrete. In 2004, Yueping Jiang [1] generalized Theorem J and proved the following.Theorem Y ([1, Theorem 2]) : Let G ˆ M (R n ) be non-elementary. Then G isdiscrete if and only if W ( G ) is discrete and each two-generator non-elementarysubgroup is discrete. Here W ( G ) = ff 2 G : f xes every limit point of Gg .Jiang also gave an example to show that the condition that \

Let {, } be a sequence of i.i.d. nondegenerate random variables which is in the domain of attraction of the normal law with mean zero and possibly infinite variance. Denote , and . Then for d > -1, we showed that under some regularity conditions, holds in this paper, where If g denotes the indicator function.

We improve some inequalities of Ostrowski-like type and further generalize them.Key words: Inequality, Error, Integral, Taylor, Ostrowski2000 MSC: 26D10, 41A55, 65D301. IntroductionIn 1938, Ostrowski [8] proved the following interesting integral inequality which has received con-siderable attention from many researchers.Theorem 1 (See [8]). Let f : [a,b] → R be continuous on [a,b] and differentiable on (a,b) whosederivative function f 0 : (a,b) → R is bounded on (a,b), i.e., kf 0 k ∞ = sup t∈(a,b) |f 0 (t)| < ∞. ThenZb f(x)−1b−a af(t)dt ∞5 14+x− a+b22 (b−a) 2 !(b−a)kf 0 k (1)for all x ∈ [a,b].This inequality gives an upper bound for the approximation of the integral average 1b−a R ba f(t)dtby the value f(x) at point x ∈ [a,b]. The first generalization of Ostrowskis inequality was given byG.V. Milovanovi´c and J.E. Peˇcari´c in [7]. However, note that estimate (1) can be applied only if f 0 is bounded. In the first part of this paper, we will improve (1) by assuming f 0 ∈ L p

In this paper we give a new characterization of real hyper- surfaces of type B, that is, a tube over a totally geodesicQP n in complex two-plane Grassmannians G2(C m+2 ), where m = 2n, with the Reeb vec- tor » belonging to the distribution D, where D denotes a subdistribution in the tangent space TxM such that TxM = D'D? for any point x 2 M and D? = Span{»1,»2,»3 }.

If f is M-harmonic and integrable with respect to a weighted radial measure "fi over the unit ball Bn of C n , then R Bn (f - ˆ) d"fi = f(ˆ(0)) for every ˆ 2 Aut(Bn). Equivalently f is fixed by the weighted Berezin transform; Tfif = f. In this paper, we show that if a function f defined on Bn satisfies R(f - `) 2 L 1 (Bn) for every ` 2 Aut(Bn) and Sf = rf for some |r| = 1, where S is any convex combination of the iterations of Tfi 0 s, then f is M-harmonic.

In this paper, we shall show that for any entire function f, the function of the form f m (f n i 1)f 0 has no non-zero finite Picard value for all positive integers m, n 2 N possibly except for the special case m = n = 1. Furthermore, we shall also show that for any two non- constant meromorphic functions f and g, if f m (f n i1)f 0 and g m (g n i1)g 0 share the value 1 weakly, then f · g provided that m and n satisfy some conditions. In particular, if f and g are entire, then the restrictions on m and n could be greatly reduced.

In this paper, we shall prove several results concerning Ricci curvature of a Riemannian manifold (M,g) := (SU(2),g) with an arbi- trary given left invariant metric g. First of all, we obtain the maximum (resp. minimum) of {r(X) := Ric(X,X) | ||X||g = 1,X 2 X(M)}, where Ric is the Ricci tensor field on (M,g), and then get a necessary and sucient condition for the Levi- Civita connection r on the manifold (M,g) to be projectively flat. Fur- thermore, we obtain a necessary and sucient condition for the Ricci curvature r(X) to be always positive (resp. negative), independently of the choice of unit vector field X.

Let m be a positive integer. We show that for any given real number ∈ (0; 1) and complex number with | | ≤ 1 which satisfy e2i m 1, there exists a Blaschke product B of degree 2m + 1 which has a fixed point of multiplier m at the point at infinity such that the restriction of the Blaschke product B on the unit circle is a critical circle map with rotation number . Moreover if the given real number is irrational of bounded type, then a modified Blaschke product of B is quasiconformally conjugate to some rational function of degree m + 1 which has a fixed point of multiplier m at the point at infinity and a Siegel disk whose boundary is a quasicircle containing its critical point. point of f and λ = f ' (z0) is called the multiplier of z0 if z0 ∈ C. The multiplier of z0 = ∞ is defined as the multiplier of the origin for ψ ◦ f ◦ ψ −1 , where ψ(z) = 1/z. The fixed point z0 is attracting, repelling or indifferent if its multiplier λ satisfies that |λ| 1 or |λ| = 1 respectively. Attracting fixed points belong to the Fatou set and repelling fixed points belong to the Julia set. In the case that z0 is indifferent, the classification is more complicated. The fixed point z0 is parabolic, a Siegel point or a Cremer point if its multiplier is a root of unity, z0 ∈ F (f ) or z0 ∈ J(f ) respectively. Parabolic fixed points belong to the Julia set. The Fatou component containing a Siegel point is called a Siegel disk centered at z0. Non-repelling fixed points "capture" at least one critical point of f , which is a solution of the equation f ' (z) = 0. In this paper, we investigate rational functions with Siegel disks. Let f

It is shown that α-regular McShane and strong Perron integrals considered in [1] are equivalent to the usual McShane integral. This note is related to the recent paper [1], where several Perron and McShane-type integrals in R are introduced and compared. We show here that all those integrals are in fact equivalent to the usual McShane integral. We borrow all the notation and terminology from [1]. The n-dimensional interval I = [a1, b1]× · · · × [an, bn] is said to be α-regular, α ∈ (0, 1), if r(I) = mini(bi − ai) maxi(bi − ai) > α. Let I0 be a fixed interval in R and I the family of all subintervals of I0. With F we denote the free full interval basis F = {(I, x) : I ∈ I, x ∈ I0}. For a given function δ : I0 → (0,∞), called a gauge, and a given α ∈ (0, 1) we define Fδ = { (I, x) ∈ F : I ⊂ U(x, δ(x))}, Fα δ = { (I, x) ∈ F : r(I) > α, I ⊂ U(x, δ(x))}. A finite subset P of Fδ (of Fα δ ) is called an Fδ-division (an Fα δ -division respectively) if for distinct pairs (I1, x1) and (I2, x2) in P, the intervals I1 and I2 are nonoverlapping. If, moreover, ⋃ (I,x)∈P I = I0, then P is called respectively an Fδ-partition and an Fα δ -partition of I0. We recall the definition of the McShane integral. Definition 1. A point function f on I0 is McShane integrable (M-integrable, in brief), with the integral A, if for each 2 > 0 there exists a gauge δ such that ∣∣ ∑ (I,x)∈π f(x)|I| −A ∣∣ < 2 for every Fδ-partition π of I0. Received December 19, 2005. 2000 Mathematics Subject Classification. 26A39.

In this paper, we will deflne the Bergman projection type operator Pr and flnd conditions on which the operator Pr is bound-ed on L p (B;d"). By using the properties of the Bergman projection type operator Pr, we will show that if f 2 L p(B;d"), then (1i k w k 2 )rf(w) ¢ z 2 L p (B;d"). We will also show that if (1i k w k 2 ) rf(w)¢z hz;wi 2 L p (B;d"), then f 2 L p(B;d").

In this paper, we investigate some properties of the entire function of the hyper order less than sharing one value CM with its k-th derivative.

In this paper we seek a positive, radially symmetric and energy minimizing solution of an m-Laplacian equation, -div$({\nabla}u^{m-2}{\nabla}u)\;=\;h(u)$. In the variational sense, the solutions are the critical points of the associated functional called the energy, $J(v)\;=\;\frac{1}{m}\;\int_{R^N}\;{\nabla}v^m\;-\;\int_{R^N}\;H(v)dx,\;where\;H(v)\;=\;{\int_0}^v\;h(t)dt$. A positive, radially symmetric critical point of J can be obtained by solving the constrained minimization problem; minimize{$\int_{R^N}{\nabla}u^mdx\;\int_{R^N}\;H(u)d;=\;1$}. Moreover, the solution minimizes J(v).

In (11) Kim obtained fixed point theorems for maps defined on some "locally G-convex" subsets of a generalized convex space. Theorem 2 in Kim's article determines us to introduce, in this paper, the notion of K-G-convex space. In this framework we obtain fixed point theorems, section properties and minimax inequalities.