In the present investigation, by applying two different normalizations of the Jackson and Hahn-Exton $q$-Bessel functions tight lower and upper bounds for the radii of convexity of the same functions are obtained. In addition, it was shown that these radii obtained are solutions of some transcendental equations. The known Euler-Rayleigh inequalities are intensively used in the proof of main results. Also, the Laguerre-Pólya class of real entire functions plays an important role in this work.
The radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1$, have been determined for normalizations of $q$-Bessel function, Bessel function of first kind of order $ν$, Lommel function of first kind and Legendre polynomial of odd degree.
The main purpose of this paper is to determine the radii of starlikeness and convexity of the generalized $\emph{k}-$Bessel functions for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class plays an crucial role in this paper. Moreover, the interlacing properties of the zeros of $\emph{k}-$Bessel function and its derivative is also useful in the proof of the main results. By making use of the Euler-Rayleigh in…
The main purpose of the present paper is to ascertain the radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1,$ of normalized hyper-Bessel function.
Sufficient conditions on associated parameters $p,b$ and $c$ are obtained so that the generalized and \textquotedblleft{normalized}\textquotedblright{} Bessel function $u_p(z)=u_{p,b,c}(z)$ satisfies $|(1+(zu''_p(z)/u'_p(z)))^2-1|<1$ or $|((zu_p(z))'/u_p(z))^2-1|<1$. We also determine the condition on these parameters so that $-(4(p+(b+1)/2)/c)u'_p(z)\prec\sqrt{1+z}$. Relations between the parameters $μ$ and $p$ are obtained such that the normalized Lommel function of first kind $h_{μ,p}(z)$ satisfies the subordination $1+(zh''_{μ,p}(z)/h'_{μ,p}(z))\prec\sqrt{1+z}$. Moreover, the properties of…
Some geometric properties of a normalized hyper-Bessel functions are investigated. Especially we focus on the radii of starlikeness, convexity, and uniform convexity of hyper-Bessel functions and we show that the obtained radii satisfy some transcendental equations. In addition, we give some bounds for the first positive zero of normalized hyper-Bessel functions, Redheffer-type inequalities, and bounds for this function. In this study we take advantage of Euler-Rayleigh inequalities and Laguerre-Pólya class of real entire functions, intensively.
In this investigation our main aim is to determine the radius of uniform convexity of the some normalized q-Bessel and Wright functions. Here we consider six different normalized forms of q-Bessel functions, while we apply three different kinds of normalizations of Wright function. Also, we have shown that the obtained radii are the smallest positive roots of some functional equations.
In this paper our aim is to find the radii of starlikeness and convexity of Bessel function derivatives for three different kind of normalization. The key tools in the proof of our main results are the Mittag-Leffler expansion for nth derivative of Bessel function and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized nth derivative of Bessel function. The main results of the paper are natural extensions of some known results on classical B…
The asymptotic behaviour, with respect to the large order, of the radii of starlikeness of two types of normalised Bessel functions is considered. We derive complete asymptotic expansions for the radii of starlikeness and provide recurrence relations for the coefficients of these expansions. The proofs rely on the notion of Rayleigh sums and asymptotic inversion. The techniques employed in the paper could be useful to treat similar problems where inversion of asymptotic expansions is involved.
In this paper, it is aimed to determine the radii of starlikeness and convexity of the normalized generalized Struve functions for three different kinds of normalization and to find tight lower and upper bounds for the radius of starlikeness and convexity of these normalized Struve functions by making use of Euler-Rayleigh inequalities. The Laguerre-Pólya class of entire functions has a crucial role in constructing our main results.
In this paper, we deal with some geometric properties including starlikeness and convexity of order $α$ of Jackson's second and third $q$-Bessel functions which are natural extensions of classical Bessel function $J_ν$. In additon, we determine some conditions on the parameters such that Jackson's second and third $q$-Bessel functions belong to the Hardy space and to the class of bounded analytic functions.
In this paper we deal with the radii of starlikeness and convexity of the $q-$Mittag--Leffler function for three different kinds of normalization by making use of their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. By applying Euler-Rayleigh inequalities for the first positive zeros of these functions tight lower and upper bounds for the radii of starlikeness of these functions are obtained. The Laguerre-Pólya class of real entire functions plays a pivotal role in this investigation.
In this paper our aim is to find the radii of $γ$-Spirallike of order $α$ and convex $γ$-Spirallike of order $α$ for three different kinds of normalizations of the function $N_ν(z)=az^2J_ν^{\prime\prime}(z)+bzJ_ν^{\prime}(z)+cJ_ν(z),$ where $J_ν(z)$ is the Bessel function of the first kind of order $ν.$ Moreover, the $\mathcal{S}^*\left(φ\right)-$radii and $\mathcal{C}\left(φ\right)-$radii of these normalized functions are investigated. The tables are created and visual verification with graphs are made by giving special values to the real numbers $a,$ $b$ and $c$ in the obtained results.
The object of the present paper is to study of radius of convexity two certain integral operators as follows \begin{equation*} F(z):=\int_{0}^{z}\prod_{i=1}^{n}\left(f'_i(t)\right)^{γ_i}{\rm d}t \end{equation*} and \begin{equation*} J(z):=\int_{0}^{z}\prod_{i=1}^{n}\left(f'_i(t)\right)^{γ_i}\prod_{j=1}^{m} \left(\frac{g_j(z)}{z}\right)^{λ_j}{\rm d}t, \end{equation*} where $γ_i, λ_i\in\mathbb{C}$, $f_i$ $(1\leq i\leq n)$ and $g_j$ $(1\leq j\leq m)$ belong to the certain subclass of analytic functions.
In this paper our aim is to find the radii of starlikeness and convexity of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic in the unit disk of the complex plane. The characterization of entire functions from Laguerre-Pólya class and a result of H. Kumar and M.A. Pathan on the reality of the zeros of generalized Mittag-Leffler functions, which origins goes back to Dzhrbashyan, Ostrovskiĭ and Peresyolkova, play important roles in this paper. Moreover, the interlacing …
In this paper, four parameters Wright function is considered. Certain geometric properties such as starlikeness, convexity, uniform convexity and close-to-convexity are discussed for this function. Further, certain geometric properties of normalized Bessel function of the first kind and two parameters Wright function are studied as a consequence. Interesting corollaries and examples are provided to support that these results are better than the existing ones and improve several results available in the literature.
In this paper our aim is to investigate the radii of $η-$uniformly convexity, $α-$convexity, $η-$parabolic starlikeness and strong starlikeness of order $ρ$ of the generalized Mittag-Leffler function for three different kinds of normalization by using their Hadamard factorization in such a way that the resulting functions are analytic. The basic tool of this study is the Mittag-Leffler function in series. Also we have shown that the obtained radii are the smallest positive roots of some functional equations.
By using Rafid operator we define the subclass $R_{μ,p}^δ(α;A,B)$ and $P_{μ,p}^δ(α; A,B)$ of analytic and p-valent functions with negative coefficients we investigate some sharp results including coefficients estimates, distortion theorem, radii of starlikeness, convexity, close-to-convexity, and modified-Hadamard product. Finally, we give an application of fractional calculus and Bernadi-Libora-Livingstion operator.
In this paper our aim is to find the radii of starlikeness and convexity for three different kind of normalization of the $N_ν(z)=az^{2}J_{ν}^{\prime \prime }(z)+bzJ_{ν}^{\prime}(z)+cJ_{ν}(z)$ function, where $J_ν(z)$ is called the Bessel function of the first kind of order $ν.$ The key tools in the proof of our main results are the Mittag-Leffler expansion for $N_ν(z)$ function and properties of real zeros of it. In addition, by using the Euler-Rayleigh inequalities we obtain some tight lower and upper bounds for the radii of starlikeness and convexity of order zero for the normalized $N_ν(z)…
The main object of the present paper is to, introduce the. class of meromorphic univalent functions Involving! hypergeomatrc function .We obtain~ some interesting geometric properties according to coefficient inequality , growth and distortion theorems , radii of starlikeness and convexity for the" functions in our subclass.
In the present paper, we discuss several basic properties of a class of quasiconformal close-to-convex harmonic mappings with starlike analytic part, such results as coefficient inequalities, an integral representation, a growth theorem, an area theorem, and radii of close-to-convexity of partial sums of the class, are derived.
The main purpose of the present paper is to determine the radii of starlikeness and convexity associated with lemniscate of Bernoulli and the Janowski function, $(1+Az)/(1+Bz)$ for $-1\leq B<A\leq 1,$ of normalized Wright functions. The key tools in the proof of our main results are the infinite product representation of Wright function and properties of real zeros of the Wright function and its derivative.
