Basic and fractional q-calculus and associated Fekete-Szegő problem for p-valently q-starlike functions and p-valently q-convex functions of complex order

Abstract

In this paper, we introduce and study some subclasses of p-valently analytic functions in the open unit disk U by applying the q-derivative operator and the fractional q-derivative operator in conjunction with the principle of subordination between analytic functions. For the Taylor-Maclaurin coefficients fa k g 1 kDpC1 of each of these subclasses of p-valently analytic functions, we derive sharp bounds for the Fekete-Szegő functional given by a pC2 a 2 pC1ˇ: Relevant connections of the results presented in this paper with those derived in earlier works are also considered.

An Upper Bound of the Third Hankel Determinant for a Subclass of q-Starlike Functions Associated with k-Fibonacci Numbers

“…Further developments based upon the the q-calculus can be motivated by several recent works which are reported in (for example) [24,25], which dealt essentially with the second and the third Hankel determinants, as well as [26][27][28][29], which studied many different aspects of the Fekete-Szegö problem.…”

Some subordinating results for classes of functions defined by Sǎlǎgean type q derivative operator

“…. ( see [15], [1], , [36] [39]) For q ∈ (0, 1), the q-derivative of f ∈ A, is given by ( see [20] [21], [22], [27], [28], [30], [31], [32], [34], [36], [37], [38], [40] and [41])…”

Some properties of analytic functions associated with fractional q-calculus operators

“…For this purpose, we refer the reader to the various definitions, notations and conventions, which are considerably detailed in our earlier paper (see, for details, [22]; see also [8]). For a fixed 2 C, a set D is called a -geometric set if and only if both´2 D and ´2 D. For a function f defined on a q-geometric set, we make use of Jackson's q-derivative and q-integral .0 < q < 1/ of a function on a subset of C, which are already introduced in several earlier investigations (see, for example, [2], [4], [6], [8], [9], [10], [14], [15], [16], [17], [21], [22] and [25]). Now, for a complex-valued function f .´/; we introduce the fractional q-calculus operators as follows (see, for example, [12] and [13]; see also [1]).…”