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Using the Hadamard factorization, the exponential radii of starlikeness and convexity for various special functions like Wright function, Lommel function, Struve function, Ramanujan type entire function, cross product and product of Bessel function have been investigated. For certain ranges of the parameters appearing in these special functions, the precise values of the exponential radii of starlikeness and convexity are calculated as the solutions of transcendental equations. The interlacing property of the zeros of special functions and their derivatives is the fundamental technique utilized to demonstrate these results. [ABSTRACT FROM AUTHOR]

In the paper, the authors introduce two notions, the normalized remainders, or say, the normalized tails, of the Maclaurin power series expansions of the sine and cosine functions, derive two integral representations of the normalized tails, prove the nonnegativity, positivity, decreasing property, and concavity of the normalized tails, compute several special values of the Young function, the Lommel function, and a generalized hypergeometric function, recover two inequalities for the tails of the Maclaurin power series expansions of the sine and cosine functions, propose three open problems about the nonnegativity, positivity, decreasing property, and concavity of a newly introduced function which is a generalization of the normalized tails of the Maclaurin power series expansions of the sine and cosine functions. These results are related to the Riemann–Liouville fractional integrals. [ABSTRACT FROM AUTHOR]

The authors of (Complex Var Elliptic Equ 54(7):689–696, 2009) proved a kind of monotony of Bessel functions using integral represen- tations. This paper establishes a monotonicity property for the normalized Lommel function of the first kind using the method of subordination factor sequences. [ABSTRACT FROM AUTHOR]

Our purpose in the present investigation is to study certain geometric properties such as the close-to-convexity, convexity, and starlikeness of the hypergeometric function z 1 F 2 (a ; b , c ; z) in the open unit disk. The usefulness of the main results for some familiar special functions like the modified Sturve function, the modified Lommel function, the modified Bessel function, and the F 1 0 (− ; c ; z) function are also mentioned. We further consider a boundedness property of the function F 2 1 (a ; b , c ; z) in the Hardy space of analytic functions. Several corollaries and special cases of the main results are also pointed out. [ABSTRACT FROM AUTHOR]

Using the admissibility condition, we obtain certain third order differential subordination results for an analytic function p with p(0) = 1 belonging to the class of Janowski functions, P[A,B]. As an application, certain second order differential inequalities involving special functions are obtained. [ABSTRACT FROM AUTHOR]

In this paper, certain new classes of analytic functions are introduced by using convolution with normalized Lommel function and the concept of bounded radius rotation. Several inclusion relationships and some new geometric properties of these functions are established. [ABSTRACT FROM AUTHOR]

Sufficient conditions are obtained on the parameters of Lommel function of the first kind, generalized Struve function of the first kind, and the confluent hypergeometric function under which these special functions become exponential convex and exponential starlike in the open unit disk. The method of differential subordination is employed in proving the results. Few examples are also provided to illustrate the results obtained. [ABSTRACT FROM AUTHOR]

A normalized analytic function f is lemniscate starlike if the quantity z f ′ (z) / f (z) lies in the region bounded by the right half of the lemniscate of Bernoulli | w 2 - 1 | = 1 . It is Janowski starlike if the quantity z f ′ (z) / f (z) lies in the disk whose diametric end points are (1 - A) / (1 - B) and (1 + A) / (1 + B) for - 1 ≤ B < A ≤ 1 . The radii of lemniscate starlikeness and Janowski starlikeness have been determined for normalizations of q-Bessel functions, Bessel functions of first kind of order ν and Lommel functions of first kind. Corresponding convexity radii are also determined. [ABSTRACT FROM AUTHOR]

Sufficient conditions on associated parameters p, b and c are obtained so that the generalized and "normalized" Bessel function up(z) = up,b,c(z) satisfies the inequalities ∣(1 + (zu″p(z)/u′p(z)))2 − 1∣ < 1 or ∣((zup(z))′/up(z))2 − 1∣ < 1. We also determine the condition on these parameters so that − (4 (p + (b + 1) / 2) / c) u p ' (x) ≺ 1 + z . Relations between the parameters μ and p are obtained such that the normalized Lommel function of first kind hμ,p(z) satisfies the subordination 1 + (z h μ , p ' ' (z) / h μ , q ' (z)) ≺ 1 + z . Moreover, the properties of Alexander transform of the function hμ,p(z) are discussed. [ABSTRACT FROM AUTHOR]

In the present paper, we consider a class of second-order singular differential operators which generalize the well-known Bessel differential operator. The associated eigenfunctions are the Bessel–Wright functions. These functions can be obtained by the action of the Riemann–Liouville operator on the normalized Bessel functions. We introduce a Bessel–Wright transform with Bessel–Wright functions as kernel which is connected to the classical Bessel–Fourier transform via the dual of the Riemann–Liouville operator. The Bessel–Wright transform leaves invariant the Schwartz space and sends the set of functions indefinitely differentiable with compact support into the Paley–Wiener space. We conclude the paper by proving two variants of the inversion formulas. [ABSTRACT FROM AUTHOR]

In the paper, the analytical forms of fractional order derivatives of sinusoidal function according to definitions of Riemann - Liouville and Caputo are presented. To determine the analytical form of the integrals appearing in definitions of derivatives of fractional order the Lommel functions from hypergeometrical functions family were applied. With the use of properties of the derivatives of fractional order - differential-integral there were presented the conception of generalized element of a single equation, which depending on the value of the derivative order, can be inductor, resistor, capacitor, or a hypothetical element of a fractional order differential equation. [ABSTRACT FROM AUTHOR]

The aim of this paper is to investigate the known large argument asymptotic series of the Lommel function by Stieltjes transform representations. We obtain a number of properties of this asymptotic expansion, including explicit and realistic error bounds, and an exponentially improved asymptotic expansion. An interesting consequence related to the large argument asymptotic series of the Struve function is also proved. [ABSTRACT FROM AUTHOR]