Akshaya Kumar Mishra

Certain subclasses of analytic and bi-univalent functions

In the present paper, we introduce and investigate two interesting subclasses of normalized analytic and univalent functions in the open unit disk U≔{z:z∈Cand|z|<1}, whose inverse has univalently analytic continuation to U. Among other results, bounds for the Taylor–Maclaurin coefficients |a2| and |a3| are found in our investigation.

Applications of fractional calculus to parabolic starlike and uniformly convex functions

Let A be the class of analytic functions in the open unit disk U. Given 0 ≤ λ < 1, let Ωλ be the operator defined on A by (ωλf)(z)=Γ(2-λ)zλDλzf(z) , where Dλzf is the fractional derivative of f of order λ. A function f in A is said to be in the class SPλ if Ωλf is a parabolic starlike function. In this paper, several basic properties and characteristics of the class SPλ are investigated. These include subordination, inclusion, and growth theorems, class-preserving operators, Fekete-Szegő problems, and sharp estimates for the first few coefficients of the inverse function.

The fekete-szegö-problem for a subclass of close-to-convex functions

Let C 1(β) be the class of normalized functions f which are analytic in the open unit disk u, given by the power series: and satisfy the inequality: for some normalized univalent and convex function φ. In this paper we solve the Fekete-Szego problem for the family: by proving that

Classes of multivalent analytic functions involving the Dziok-Srivastava operator

The object of the present paper is to investigate some inclusion relationships and a number of other useful properties of several subclasses of multivalent analytic functions, which are defined here by using the Dziok-Srivastava operator. Relevant connections of the results presented here with those obtained in earlier works are pointed out.

Second Hankel Determinant for a Class of Analytic Functions Defined by Fractional Derivative

By making use of the fractional differential operator due to Owa and Srivastava, a class of analytic functions is introduced. The sharp bound for the nonlinear functional is found. Several basic properties such as inclusion, subordination, integral transform, Hadamard product are also studied.

Convex hulls and extreme points of some classes of multivalent functions

are satisfied in p q and 0 <a < 1. We denote the class S(p, q, 0) by S(p, q). The functions in S(p, q, a) are p-valent; for S(p, q, a) c S( p, q), the class of p-valent starlike functions studied by Goodman [5]. A function f(z), given by (l.l), is said to be in the class C(p, q, a) if and only if zf’(z)/q is in S(p, q, a). The class C(p, q, a) is contained in C(p, q, 0) = C(p, q), the class of p-valent convex functions [5 1. Let K( p. p, a) be the class of functionsf(z), analytic in D and of the form such that there exists a function g(z) in S(p, p, a) and a complex number E, 1 E ] = I, satisfying

Invariance of some subclass of multivalent functions under a differintegral operator

Let 𝒜 p denote the class of functions analytic in the open unit disc 𝒰 = {z ∈ ℂ : |z| < 1} and given by the series For f ∈ 𝒜 p , let A function f ∈ 𝒜 p is in the class if is satisfied, where ≺ stands for subordination. The main result of this article is the invariance of the class under the operator when μ varies. Other results include Strohäcker–Marx-like inequalities involving the operator .

Some applications of Schwarz Lemma for operators

A generalized Schwarz lemma and some Harnack type inequalities for operators have been obtained in this paper.

Coefficients of multivalent symmetric functions of bounded boundary rotation

In this paper the sharp coefficient estimate problem for the classesCp(β, m) andVp(k,m) of multivalent close-to-convex functions of order β and multivalent functions of bounded boundary rotation of at mostkπ, whose functions are given bym-fold symmetric gap series, have been discussed respectively for β≥1−p/m>0 andk≥2(m/p). Moreover, it is shown that every function inVp(k,m) arep-valent close-to-convex; hencep-valent; ifk<2 (1+m/p).

A generalization of the Srivastava–Attiya transform and associated classes

In this paper, by making use of a generalization of the Srivastava–Attiya operator, a new class of multivalent analytic functions is introduced and investigated. Under suitable constraints on relevant parameters, a result on the invariance of this class under a related integral operator is obtained; which, furthermore, yields an inclusion theorem. Sharp results on the converse problem are also presented.