Ravinder Krishna Raina

Some inclusion properties for certain subclasses of strongly starlike and strongly convex functions involving a family of fractional integral operators

A known family of fractional integral operators (with the Gauss hypergeometric function in the kernel) is used here to define some new subclasses of strongly starlike and strongly convex functions of order β and type α in the open unit disk 𝕌. For each of these new function classes, several inclusion relationships associated with the fractional integral operators are established. Some interesting corollaries and consequences of the main inclusion relationships are also considered.

On α-convex functions related to shell-like functions connected with Fibonacci numbers

Abstract This paper presents a new class SLM α of functions f(z) analytic and normalized in the open unit disc U = { z : | z | 1 } (which is related to a shell-like curve and associated with Fibonacci numbers) satisfying the condition that α 1 + zf ″ ( z ) f ′ ( z ) + ( 1 - α ) zf ′ ( z ) f ( z ) ∈ p ˜ ( U ) ( z ∈ U ) , where α is a real number and p ˜ ( z ) = τ z + τ 2 z 2 1 - τ z - τ 2 z 2 ( τ = ( 1 - 5 ) / 2 ; z ∈ U ) . The class SLM α being closely related to the classes of starlike and convex functions, we apply some basic techniques to investigate certain interesting properties (given below) for this class of functions. Some important observations of the main results are also mentioned.

A unified presentation of certain subclasses of prestarlike functions with negative coefficients

Abstract The main object of this paper is to introduce and investigate various properties and characteristics of a unified class P(α,β,σ), of prestarlike functions with negative coefficients. The results presented here involve distortion inequalities and modified Hadamard products (or convolution) of functions belonging to the class P(α,β,σ). Growth and distortion theorems involving fractional integrals and fractional derivatives are also considered. Relevant connections of some of these results with those given in earlier works are briefly pointed out.

Solution of Abel-type integral equation involving the Appell hypergeometric function

The present paper gives a formal solution of a certain Abel-type integral equation involving the Appell hypergeometric function in the kernel. The integral equation and its solution give rise to new forms of generalized fractional calculus operators (viz. the generalized fractional integrals and generalized fractional derivatives). These and their various consequences are also mentioned. The concluding remarks briefly point out possibilities of further work concerning the operators studied in this paper.

Harmonic univalent functions associated with Wright's generalized hypergeometric functions

The purpose of this paper is to study certain harmonic univalent mappings involving the Wrights generalized hypergeometric functions. We investigate the usual characteristics associated with such harmonic mappings and also mention their validity conditions. As consequences of our main results, we deduce several results involving the generalized Mittag–Leffler function, the Bessel–Maitland function and the generalized hypergeometric function.

Some subclasses of analytic functions associated with fractional calculus operators

Abstract This paper begins by studying several properties and characteristics of certain subclasses of analytic functions with positive coefficients. The results investigated here include various coefficient inequalities, distortion properties, and the radii of close-to-convexity. Inclusion theorems involving the Hardy space of analytic functions and the class of functions whose derivative has a positive real part are also investigated. Relationships between certain subclasses of analytic functions (involving fractional derivative operators) with negative coefficients and a certain generalized fractional integral operator are then studied. Various known or new special cases of our results are also pointed out.

A class of numbers associated with the Lucas numbers

The main object of this paper is to present a systematic investigation of a new class of numbers associated with the familiar Lucas numbers. The various results obtained here for this class of numbers include explicit hypergeometric representations, generating functions, recurrence relations, and summation formulas.

On a class of starlike functions related to a shell-like curve connected with Fibonacci numbers

In this paper we investigate an interesting subclass SL of analytic univalent functions in the open unit disc on the complex plane. This class was introduced by Sokol [J. Sokol, On starlike functions connected with Fibonacci numbers, Folia Scient. Univ. Tech. Resoviensis 175 (1999) 111-116]. The class SL is strongly related to the class KSL considered earlier by the authors of the present work in their paper [J. Dziok, R. K. Raina, J. Sokol, Certain results for a class of convex functions related to shell-like curve connected with Fibonacci Numbers, Comput. Math. Appl. 61 (2011) 2606-2613]. Apart from furnishing some genuine remarks, we present certain new results for the class SL of functions, and also mention some relevant cases for this function class.

Differential subordinations and α-convex functions

This article presents some new results on the class SLMα of functions that are analytic in the open unit disc U = {z:|z|<1} satisfying the conditions that f(0)=0,f′(0)=1,andα(1+zf′′(z)f′(z))+(1−α)zf′(z)f(z)∈p˜(u) for all z ∈ U, where α is a real number and p˜(z)=1+τ2z21−τz−τ2z2(z∈u) . The number τ=(1−5)/2 is such that τ2 = 1 + τ. The class SLMα introduced by J. Dziok, R.K. Raina, and J. Sokol [3, Appl. Math. Comput. 218 (2011), 996–1002] is closely related to the classes of starlike and convex functions. The article deals with several ideas and techniques used in geometric function theory and differential subordinations theory.

Extremal problems in the class of analytic functions associated with Wright's function

Abstract Making use of the extremal points theory, we investigate some extremal problems for the class of analytic functions with negative coefficients defined by Wrights generalized hypergeometric function.