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In this paper, some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product, three different normalizations are investigated and for each of the six functions, the radii of starlikeness and convexity are precisely determined by using their Hadamard factorization. Necessary and sufficient conditions are also given for the parameters such that the six normalized functions are starlike in the open unit disk; however, the convex case is open for further research. The characterization of entire functions from the Laguerre–Polya class via hyperbolic polynomials plays an important role in this paper. Moreover, the interlacing properties of the zeros of the cross-product and product of Bessel functions and their derivatives are also useful in the proof of the main results.

Tight lower and upper bounds for the radius of univalence of some normalized Bessel, Struve and Lommel functions of the first kind are obtained via Euler-Rayleigh inequalities. It is shown also that the radius of univalence of the Struve functions is greater than the corresponding radius of univalence of Bessel functions. Moreover, by using the idea of Kreyszig and Todd, and Wilf it is proved that the radii of univalence of some normalized Struve and Lommel functions are exactly the radii of starlikeness of the same functions. The Laguerre-P\'olya class of entire functions plays an important role in our study., Comment: 12 pages

The reality of the zeros of the product and cross-product of Bessel and modified Bessel functions of the first kind is studied. As a consequence the reality of the zeros of two hypergeometric polynomials is obtained together with the number of the Fourier critical points of the normalized forms of the product and cross-product of Bessel functions. As an application some geometric properties of the normalized forms of the cross-product and product of Bessel and modified Bessel functions of the first kind are studied. For the cross-product and the product three different kind of normalization are considered and for each of the six functions tight lower and upper bounds are given for the radii of starlikeness and convexity, via Euler–Rayleigh inequalities. Necessary and sufficient conditions are also given for the parameters such that four from the six normalized functions are convex in the open unit disk.

In this paper necessary and sufficient conditions are deduced for the close-to-convexity of some special combinations of Bessel functions of the first kind and their derivatives by using a result of Shah and Trimble about transcendental entire functions with univalent derivatives and some newly discovered Mittag–Leffler expansions for Bessel functions of the first kind.

In this paper our aim is to deduce some new results on the zeros of the derivatives of some Lommel and Struve functions of the first kind and to apply these results in order to determine the radii of convexity of some normalized Lommel and Struve functions of the first kind.

In this work we present some geometric properties (like star-likeness and convexity of order α and also close-to-convexity of order(1+α)/2) for normalized of Lommel functions of the first kind. In orderto prove our main results, we use the technique of differential subordina-tions and some inequalities. Furthermore, we present some applicationsof convexity involving Lommel functions associated with the Hardy spaceof analytic functions, i.e., we obtain conditions for the function h µ,v (z)to belong to the Hardy space H p . 1. IntroductionLet H be the set of all analytic functions on the open unit disk U ={z: |z

Let k − ST [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the class of normalized analytic functions defined in the open unit disk satisfying ℜ ( ( B − 1 ) z f ′ ( z ) f ( z ) − ( A − 1 ) ( B + 1 ) z f ′ ( z ) f ( z ) − ( A + 1 ) ) > k | ( B − 1 ) z f ′ ( z ) f ( z ) − ( A − 1 ) ( B + 1 ) z f ′ ( z ) f ( z ) − ( A + 1 ) − 1 | . $$\Re \left( {{{(B - 1){{zf'(z)} \over {f(z)}} - (A - 1)} \over {(B + 1){{zf'(z)} \over {f(z)}} - (A + 1)}}} \right) > k\left| {{{(B - 1){{zf'(z)} \over {f(z)}} - (A - 1)} \over {(B + 1){{zf'(z)} \over {f(z)}} - (A + 1)}} - 1} \right|.$$ and let k − UCV [A, B], k ≥ 0, −1 ≤ B < A ≤ 1 be the corresponding class satisfying ℜ ( ( B − 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A − 1 ) ( B + 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A + 1 ) ) > k | ( B − 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A − 1 ) ( B + 1 ) ( z f ′ ( z ) ) ′ f ′ ( z ) − ( A + 1 ) − 1 | . $$\Re \left( {{{(B - 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A - 1)} \over {(B + 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A + 1)}}} \right) > k\left| {{{(B - 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A - 1)} \over {(B + 1){{\left( {zf'(z)} \right)^{\prime } } \over {f'(z)}} - (A + 1)}} - 1} \right|.$$ For an appropriate δ > 0, the δ neighborhood of a function f ∈ k − UCV [A, B] is shown to consist of functions in the class k − ST [A, B].

The authors obtain the Fekete-Szegő inequality (according to parameters $s$ and $t$ in the region $s^{2}+st+t^{2} 3,$ $s\neq t$ and $s+t\neq 2$) for certain normalized analytic functions $f(z)$ belonging to $k\text {\rm -UST}_{\lambda ,\mu }^{n}(s,t,\gamma )$ which satisfy the condition \begin {equation*} \Re \bigg \{ \frac {(s-t)z ( D_{\lambda ,\mu }^{n}f(z))'} {D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}\bigg \} >k \biggl \vert \frac {(s-t)z ( D_{\lambda ,\mu }^{n}f(z))'}{D_{\lambda ,\mu }^{n}f(sz)-D_{\lambda ,\mu }^{n}f(tz)}{-1} \biggr \vert +\gamma , \quad z\in \mathcal {U} . \end {equation*} Also certain applications of the main result a class of functions defined by the Hadamard product (or convolution) are given. As a special case of this result, the Fekete-Szegő inequality for a class of functions defined through fractional derivatives is obtained.

Let gp,b,c n (z )= z+ n ∑ m=1 bmz m+1 be the sequence of partial sums of generalized ∞ ∑ m=1 bmz m+1 where bm = (−c/4) m m!(κ)m and κ :=

In this paper, we present some immediate applications of convexity involving Struve functions associated with the Hardy space of analytic functions, i.e. we obtain conditions for the functionto belong to the Hardy space

In this paper, we introduce and investigate an interesting subclass N?h,p (?, ?) of analytic and bi-univalent functions in the open unit disk U. For functions belonging to the class N?h,p (?, ?), we obtain estimates on the first two Taylor-Maclaurin coefficients |a2| and |a3|. The results presented in this paper would generalize and improve some recent works of ?a?lar et al. [3], Xu et al. [10], and other authors.

Geometric properties of the classical Lommel and Struve functions, both of the first kind, are studied. For each of them, there different normalizations are applied in such a way that the resulting functions are analytic in the unit disc of the complex plane. For each of the six functions we determine the radius of starlikeness precisely., Comment: 9 pages

The aim of this paper is to define \ a new operator by using the generalized Struve functions $\sum\limits_{n=0}^{\infty }\frac{\left( -c/4\right) ^{n}}{\left( 3/2\right) _{n}\left( k\right) _{n}}z^{n+1}$ with $% k$ $=p+$ $\left( b+2\right) /2\neq 0,-1,-2,\ldots $ and $b,c,k\in \mathbb{C} $. By using this operator we define a subclass of analytic functions. We discuss some properties of this class such as inclusion problems, radius problems, and some other interesting properties related to this operator.

We give sufficient conditions for the parameters of the normalized form of the generalized Struve functions to be convex and starlike in the open unit disk.

In the present investigation, we consider two new subclasses N_{{\Sigma}}^{{\mu}}({\alpha},{\lambda}) and N_{{\Sigma}}^{{\mu}}({\beta},{\lambda}) of bi-univalent functions defined in the open unit disk U={z:|z, Comment: 7 pages, submitted to any Journal for publication