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In the paper we consider a general inequality | p n - 1 p n + 1 - p n 2 | ≤ 4 - | p 1 | 2 involving coefficients of functions with a positive real part. We prove this inequality for n = 2 and n = 3 . Consequently, the relative inequalities involving coefficients of Schwarz functions are obtained. As an application, the two sharp estimates of the Hankel determinants H 3 , 1 and H 2 , 3 are proved for functions in S ∗ (1 / 2) and M , respectively. [ABSTRACT FROM AUTHOR]

In the present article, we prove the sharp upper bounds for the second order Hankel determinants |H2(1)|,|H2(2)| and related functionals |a2a3−a4|, |a2a5−a3a4| for q-starlike functions. An upper bound for the third order Hankel determinant |H3(1)| along with the sharp upper bounds for Toeplitz determinant |Tm(n)|, where (m,n)∈{(2,2),(2,3),(3,1),(3,2)} are attained. Many known results are also obtained as corollaries of our main results.

In this study, we introduce the new subclasses, M α (sin) and M α (cos) , of α -convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of the Zalcman functional for the class M α (cos) for the case n = 3 , proving that the Zalcman conjecture holds for this value of n. Moreover, the problem of the Fekete–Szegő functional estimate for these classes is studied. [ABSTRACT FROM AUTHOR]

The purpose of the paper is to derive some interesting implications associated with some differential inequalities, for certain analytic functions in the open unit disk. Connections to previously well known results are also established. [ABSTRACT FROM AUTHOR]

Systems of non-autonomous parabolic partial differential equations over a bounded domain with nonlinear term of Carathéodory type are considered. Appropriate topologies on sets of Lipschitz Carathéodory maps are defined in order to have a continuous dependence of the mild solutions with respect to the variation of both the nonlinear term and the initial conditions, under different assumptions on the bound-maps of the nonlinearities. [ABSTRACT FROM AUTHOR]

In the present paper, we develop some implications leading to Carathéodory functions in the open disk and provide some new conditions for functions to be p-valent functions. This work also extends the findings of Nunokawa and others. [ABSTRACT FROM AUTHOR]

In this study, we introduce the new subclasses, Mα(sin) and Mα(cos), of α-convex functions associated with sine and cosine functions. First, we obtain the initial coefficient bounds for the first five coefficients of the functions that belong to these classes. Further, we determine the upper bound of the Zalcman functional for the class Mα(cos) for the case n=3, proving that the Zalcman conjecture holds for this value of n. Moreover, the problem of the Fekete–Szegő functional estimate for these classes is studied.

We define certain subclasses δ−UM(ℓ, η1, η2) and δ−UMℑ(ℓ, η1, η2) of holomorphic mappings involving some differential inequalities. These functions are actually generalizations of some basic families of starlike and convex mappings. We study sufficient conditions for f ∈ δ − UM(ℓ, η1, η2). We also discuss the characterization for f ∈ δ − UMℑ(ℓ, η1, η2) along with the coefficient bounds and other problems. Using certain conditions for functions in the class δ − UM(ℓ, η1, η2), we also define another class δ − UM* (ℓ, η1, η2) and study some subordination related result. [ABSTRACT FROM AUTHOR]

The first part of the paper is devoted to studying the continuous dependence of the solutions of Carathéodory constant delay differential equations where the vector fields satisfy classical cooperative conditions. As a consequence, when the set of considered vector fields is invariant with respect to the time-translation map, the continuity of the respective induced skew-product semiflows is obtained. These results are important for the study of the long-term behavior of the trajectories. In particular, the construction of semicontinuous semiequilibria and equilibria is extended to the context of ordinary and delay Carathéodory differential equations. Under appropriate assumptions of sublinearity, the existence of a unique continuous equilibrium, whose graph coincides with the pullback attractor for the evolution processes, is shown. The conditions under which such a solution is the forward attractor of the considered problem are outlined. Two examples of application of the developed tools are also provided. [ABSTRACT FROM AUTHOR]

Abstract In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p ( 0 ) = 1 $p(0)=1$ to satisfy Re { e i β p ( z ) } > γ $\operatorname{Re}\{ {\mathrm{e}}^{{\mathrm{i}}\beta } p(z) \} > \gamma $ or | arg { p ( z ) − γ } | < δ $| \arg \{p(z)-\gamma \} |

In this paper, we prove the sharp inequality | H 3 , 1 ⁢ (f) | ≤ 4 / 9 for starlike functions 푓, where H 3 , 1 ⁢ (f) is the third Hankel determinant, thus solving a long-standing problem. [ABSTRACT FROM AUTHOR]

In geometry, a conic is a plane curve whose coordinates satisfy a quadratic equation in two variables and can be expressed in matrix form. This equation allows deducing and expressing geometric properties of conic sections. In this article, we define certain subclasses UkS(λ, γ, τ, ρ) and U ℑ k S(λ, γ, τ, ρ) of holomorphic mappings associated with the Janowski-type mappings. These functions are actually generalizations of some basic families of starlike and convex mappings. We study sufficient conditions for f ∈ UkS(λ, γ, τ, ρ) along with the characterization, coefficient bounds and other problems. Using certain conditions for functions in the class UkS(λ, γ, ϱ, σ), we also define another class and study some subordination related result. [ABSTRACT FROM AUTHOR]

In the present paper, we develop some implications leading to Carathéodory functions in the open disk and provide some new conditions for functions to be p-valent functions. This work also extends the findings of Nunokawa and others.

In this paper, by using a technique of the first-order differential subordination, we find several sufficient conditions for an analytic function p such that p (0) = 1 to satisfy Re { e i β p (z) } > γ or | arg { p (z) − γ } | < δ for all z ∈ D , where β ∈ (− π / 2 , π / 2) , γ ∈ [ 0 , cos β) , δ ∈ (0 , 1 ] and D : = { z ∈ C : | z | < 1 } . The results obtained here will be applied to find some conditions for spirallike functions and strongly starlike functions in D . [ABSTRACT FROM AUTHOR]

Abstract In this paper, we obtain some potentially useful conditions (or criteria) for the Carathéodory functions as a certain class of analytic functions by applying Nunokawa’s lemma. We also obtain several conditions for strong starlikeness and close-to-convexity as special cases of the main results presented here.

In the present article, we prove the sharp upper bounds for the second order Hankel determinants | H 2 ( 1 ) | , | H 2 ( 2 ) | and related functionals | a 2 a 3 - a 4 | , | a 2 a 5 - a 3 a 4 | for q -starlike functions. An upper bound for the third order Hankel determinant | H 3 ( 1 ) | along with the sharp upper bounds for Toeplitz determinant | T m ( n ) | , where ( m , n ) ∈ { ( 2 , 2 ) , ( 2 , 3 ) , ( 3 , 1 ) , ( 3 , 2 ) } are attained. Many known results are also obtained as corollaries of our main results., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2024 The Author(s).)

Theory of differential subordination provides techniques to reduce differential subordination problems into verifying some simple algebraic condition called admissibility condition. We exploit the first order differential subordination theory to get several sufficient conditions for function satisfying several differential subordinations to be a Janowski function with positive real part. As applications, we obtain sufficient conditions for normalized analytic functions to be Janowski starlike functions. [ABSTRACT FROM AUTHOR]

In this paper, we obtain some potentially useful conditions (or criteria) for the Carathéodory functions as a certain class of analytic functions by applying Nunokawa's lemma. We also obtain several conditions for strong starlikeness and close-to-convexity as special cases of the main results presented here. [ABSTRACT FROM AUTHOR]

We present some sufficient conditions for an analytic function p in | z | < 1 in the complex plane to be in the class of Carathéodory functions, i.e. for Re { p (z) } > 0 in | z | < 1 . Several classes of functions may be described in terms of the class of Carathéodory functions. Therefore, these conditions may be reformulated to various sufficient conditions for starlikeness, convexity, etc. [ABSTRACT FROM AUTHOR]

In this paper, the authors derive several sufficient conditions for a function to be the Carathéodory function in the unit disk 𝔻: = {z ∈ ℂ: |z| < 1}. More precisely, for given β ∈ (–π/2, π/2), γ ∈ [0, cosβ) and δ ∈ (0, π/2], we find some sufficient conditions for an analytic function p such that p(0) = 1 to satisfy Re{e−iβp(z)} > γ or | arg {p(z)–γ} | < δ for all z ∈ 𝔻 by using the first-order differential subordination. We then apply the results obtained here in order to find some conditions for univalent functions with geometric properties such as spirallikeness and strongly starlikeness. [ABSTRACT FROM AUTHOR]

We study some already introduced and some new strong and weak topologies of integral type to provide continuous dependence on continuous initial data for the solutions of non-autonomous Carathéodory delay differential equations. As a consequence, we obtain new families of continuous skew-product semiflows generated by delay differential equations whose vector fields belong to such metric topological vector spaces of Lipschitz Carathéodory functions. Sufficient conditions for the equivalence of all or some of the considered strong or weak topologies are also given. Finally, we also provide results of continuous dependence of the solutions as well as of continuity of the skew-product semiflows generated by Carathéodory delay differential equations when the considered phase space is a Sobolev space. [ABSTRACT FROM AUTHOR]

In this paper, we found the sharp bound of the fifth coefficient of strongly starlike functions of order α with real coefficients for all α ∈ (0 , 1 ] . [ABSTRACT FROM AUTHOR]

In the present paper, we proved the sharp inequality | H 3 , 1 (f) | ≤ 1 / 9 for analytic functions f with a n : = f (n) (0) / n ! , n ∈ N , a 1 : = 1 , such that Re z f ′ (z) f (z) > 1 2 , z ∈ D : = { z ∈ C : | z | < 1 } , where H 3 , 1 (f) : = a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 is the third Hankel determinant. [ABSTRACT FROM AUTHOR]

In the present paper, the estimate of the third Hankel determinant H 3 , 1 (f) = a 1 a 2 a 3 a 2 a 3 a 4 a 3 a 4 a 5 for the class of starlike functions, i.e., for the class of analytic functions f standardly normalized such that Re (z f ′ (z) / f (z)) > 0 , z ∈ D : = { z ∈ C : | z | < 1 } , is improved. [ABSTRACT FROM AUTHOR]

Using the operator L c a defined by Carlson and Shaffer, we defined a new subclass of analytic functions ML c a ( λ ; ψ ) defined by a subordination relation to the shell shaped function ψ ( z ) = z + 1 + z 2 . We determined estimate bounds of the four coefficients of the power series expansions, we gave upper bound for the Fekete–SzegőSzegő functional and for the Hankel determinant of order two for f ∈ ML c a ( λ ; ψ ) .

The theory of (j, k)-symmetric functions has many applications in the investigation of fixed points, estimation of the absolute values of some integrals and in obtaining results of the type of Cartan’s uniqueness theorem. The concept of 2j,k-symmetric functions extends the idea of even, odd, k, 2k,  j,k-symmetric and conjugate functions. In this paper, we introduce a new class MSCPj,kα,η,δ of analytic functions using the notion of 2j,k-symmetric conjugate points. It unifies the classes SSCPj,kη,δ and CSCPj,kη,δ of starlike functions with respect to symmetric conjugate points and convex functions with respect to symmetric conjugate points, respectively. We also derive some inclusion results, integral representations and convolution conditions for functions belonging to the general function class MSCPj,kα,η,δ. The various results presented in this paper may apply to yield the corresponding (new or known) results for a number of simpler known classes. [ABSTRACT FROM AUTHOR]

In this work with a different technique we obtain upper bounds of the functional a2a4 - a23|for functions belonging to a comprehensive subclass of analytic bi-univalent functions, which is defined by subordinations in the open unit disk. Moreover, our results extend and improve some of the previously known ones. [ABSTRACT FROM AUTHOR]

We prove the sharp inequality $|H_{3,1}(f)|\leq 4/135$ for convex functions, that is, for analytic functions $f$ with $a_{n}:=f^{(n)}(0)/n!,~n\in \mathbb{N}$ , such that $$\begin{eqnarray}Re\bigg\{1+\frac{zf^{\prime \prime }(z)}{f^{\prime }(z)}\bigg\}>0\quad \text{for}~z\in \mathbb{D}:=\{z\in \mathbb{C}:|z| where $H_{3,1}(f)$ is the third Hankel determinant $$\begin{eqnarray}H_{3,1}(f):=\left|\begin{array}{@{}ccc@{}}a_{1} & a_{2} & a_{3}\\ a_{2} & a_{3} & a_{4}\\ a_{3} & a_{4} & a_{5}\end{array}\right|.\end{eqnarray}$$ [ABSTRACT FROM AUTHOR]

The well-known theory of differential subordination developed by Miller and Mocanu is applied to obtain several inclusions between Carathéodory functions and starlike functions. These inclusions provide sufficient conditions for normalized analytic functions to belong to certain class of Ma-Minda starlike functions. [ABSTRACT FROM AUTHOR]

Metric topological vector spaces of Carathéodory functions and topologies of L l o c p type are introduced, depending on a suitable set of moduli of continuity. Theorems of continuous dependence on initial data for the solutions of non-autonomous Carathéodory differential equations are proved in such new topological structures. As a consequence, new families of continuous linearized skew-product semiflows are provided in the Carathéodory spaces. [ABSTRACT FROM AUTHOR]

Let SR * be the class of starlike functions with real coefficients, i.e., the class of analytic functions f which satisfy the condition f ( 0 ) = 0 = f ′ ( 0 ) − 1 , Re { z f ′ ( z ) / f ( z ) } > 0 , for z ∈ D : = { z ∈ C : | z | < 1 } and a n : = f ( n ) ( 0 ) / n ! is real for all n ∈ N . In the present paper, it is obtained that the sharp inequalities − 4 / 9 ≤ H 3 , 1 ( f ) ≤ 3 / 9 hold for f ∈ SR * , where H 3 , 1 ( f ) is the third Hankel determinant of order 3 defined by H 3 , 1 ( f ) = a 3 ( a 2 a 4 − a 3 2 ) − a 4 ( a 4 − a 2 a 3 ) + a 5 ( a 3 − a 2 2 ) .

In this paper, we define and study some subclasses of analytic functions related with k-uniformly close-to-convex functions of higher order in the unit disc. These classes unify a number of classes previously studied. The results obtained include rate of growth of coefficients, inclusion relations, radius problems and necessary conditions for univalency. We derive many known results as special cases. [ABSTRACT FROM AUTHOR]

We propose a sufficient condition, on the nonlinear term, for the existence of solutions. This new condition is weaker than the usual sign condition and than the assumption on the existence of constant upper and lower solutions.

A pre-order and equivalence relation on the class of Hilbert space operators with positive real part are introduced, in correspondence with similar relations for contraction operators defined by Yu.L. Shmul'yan in [6]. It is shown that the pre-order, and hence the equivalence relation, is preserved by certain linear fractional transformations. As an application, the operator relations are extended to the class C(U) of Carathéodory functions on the unit disc D of C whose values are operators on a finite dimensional Hilbert space U. With respect to these relations on C(U) it turns out that the associated linear fractional transformations of C(U) preserve the equivalence relation on their natural domain of definition, but not necessarily the pre-order, paralleling similar results for Schur class functions in [3]. [ABSTRACT FROM AUTHOR]

In this paper we obtain a number of interesting relations associated with some differential inequalities in the open unit disk, U = {z : ∣z∣ < 1}. Some applications of the main results are also obtained. [ABSTRACT FROM AUTHOR]

Abstract: We consider nonlinear elliptic partial differential equations for quasilinear operators of the form subject to fully nonlinear boundary conditions involving boundary operators of the form, for each , The main goal of this paper is to give, under suitable assumptions on and , an explicit estimate for bounded solutions of these elliptic boundary value problems. Then, we establish the existence of at least one solution to such problems extending the authorʼs previous work. Our methods rely on the definition of approximate problems, deducing a priori estimates for their solutions and compactness arguments in order to pass to the limit. These methods can be applied to a large class of equations involving operators of Leray–Lions type (on suitable Banach spaces) for a general class of boundary operators which are, possibly, of the same order as . As examples, these results are shown to apply to a class of uniformly elliptic equations that occur in the theory of phase transitions, and certain elliptic systems associated with climate problems which describe the evolution of atmospheric sea-level temperatures for relatively long time scales. [Copyright &y& Elsevier]

Abstract: Scalar rational functions with a non-negative real part on the right half plane, called positive, are classical in the study of electrical networks, dissipative systems, Nevanlinna–Pick interpolation and other areas. We here study generalized positive functions, i.e. with a non-negative real part on the imaginary axis. These functions form a convex invertible cone, cic in short, and we explore two partitionings of this set: (i) into (infinitely many non-invertible) convex cones of functions with prescribed poles and zeroes in the right half plane and (ii) each generalized positive function can be written as a sum of even and odd parts. The sets of even generalized positive and odd functions form subcics. It is well known that the Nevanlinna–Pick interpolation problem is not always solvable by positive functions. Unfortunately, there is no computationally simple procedure to carry out this interpolation in the framework of generalized positive functions. Through examples it is illustrated how the two above partitionings of generalized positive functions can be exploited to introduce simple ways to carry out the Nevanlinna–Pick interpolation. Finally we show that only some of these properties are carried over to rational generalized bounded functions, mapping the imaginary axis to the unit disk. [Copyright &y& Elsevier]

Abstract: We consider the moment space corresponding to complex matrix measures defined on ( or ). We endow this set with the uniform distribution. We are mainly interested in large deviation principles (LDPs) when . First we fix an integer and study the vector of the first components of a random element of . We obtain an LDP in the set of -arrays of matrices. Then we lift a random element of into a random measure and prove an LDP at the level of random measures. We end with an LDP on Carathéodory and Schur random functions. These last functions are well connected to the above random measure. In all these problems, we take advantage of the so-called canonical moments technique by introducing new (matricial) random variables that are independent and have explicit distributions. [Copyright &y& Elsevier]

Abstract: In this contribution, we analyze the regularity conditions of a perturbation on a quasi-definite linear functional by the addition of Dirac delta functionals supported on N points of the unit circle or on its complement. We also deal with a new example of linear spectral transformation. We introduce a perturbation of a quasi-definite linear functional by the addition of the first derivative of the Dirac linear functional when its support is a point on the unit circle or two points symmetric with respect to the unit circle. Necessary and sufficient conditions for the quasi-definiteness of the new linear functional are obtained. Outer relative asymptotics for the new sequence of monic orthogonal polynomials in terms of the original ones are obtained. Finally, we prove that this linear spectral transform can be decomposed as an iteration of Christoffel and Geronimus linear transformations. [Copyright &y& Elsevier]

Abstract: Recently, by applying the principle of subordination for analytic functions in the open unit disk , Kim and Cho [I. H. Kim, N. E. Cho, Sufficient conditions for Carathéodory functions, Comput. Math. Appl. 59 (2010), 2067–2073] considered several sufficient conditions for a family of Carathéodory functions. The main purpose of this paper is to investigate some (presumably new) sufficient conditions for the class of strongly Carathéodory functions in . One illustrative example and several corollaries of the main results presented here are also considered. [Copyright &y& Elsevier]

Abstract: In this paper we discuss Weyl matrix balls in the context of the matricial versions of the classical interpolation problems named after Carathéodory and Schur. Our particular focus will be on studying the monotonicity of suitably normalized semi-radii of the corresponding Weyl matrix balls. We, furthermore, devote a fair bit of attention to characterizing the case in which equality holds for particular matricial inequalities. Solving these problems will provide us with a new perspective on the role of the central functions for the classes of Carathéodory and Schur. [Copyright &y& Elsevier]

Abstract: Let be an Hermitian linear functional defined on the linear space of Laurent polynomials. It is very well known that the Gram matrix of the associated bilinear functional in the linear space of polynomials is a Toeplitz matrix. In this contribution we analyze some linear spectral transforms of such that the corresponding Toeplitz matrix is the result of the addition of a constant in a subdiagonal of the initial Toeplitz matrix. We focus our attention in the analysis of the quasi-definite character of the perturbed linear functional as well as in the explicit expressions of the new monic orthogonal polynomial sequence in terms of the first one. [Copyright &y& Elsevier]

Abstract: In this paper, we define and study some subclasses of analytic functions by using the concept of -uniformly convexity. Several interesting properties, coefficients and radius problems are investigated. The behaviour of these classes under a certain integral operator is also studied. We indicate the relevant connections of our results with various known ones. [ABSTRACT FROM AUTHOR]

Abstract: We give a simple constructive proof of a factorization theorem for scalar rational functions with a non-negative real part on the imaginary axis. A Mathematica program, performing this factorization, is provided. [ABSTRACT FROM AUTHOR]