Hankel Determinant Problem for q-strongly Close-to-Convex Functions

Abstract

In this paper, we introduce a new class $K_{q}(\alpha), \quad 0<\alpha \leq1, \quad 0<q<1, $ of normalized analytic functions $f $ such that $\big|\arg\frac{D_qf(z)}{D_qg(z)}\big| \leq \alpha \frac{\pi}{2},$ where $g$ is convex univalent in $E= \{z: |z|<1\} $ and $D_qf $ is the $q$-derivative of $f $ defined as: $$D_qf(z)= \frac{f(z)-f(qz)}{(1-q)z}, \quad z\neq0\quad D_qf(0)= f^{\prime}(0). $$ The problem of growth of the Hankel determinant $H_n(k) $ for the class $K_q(\alpha) $ is investigated. Some known interesting results are pointed out as applications of the main results.

Second Hankel Determinant for the Subclass of Bi-Univalent Functions Using q-Chebyshev Polynomial and Hohlov Operator

“…Several writers, notably Noor [32], have investigated this determinant, with topics ranging from the rate of development of H m (j) (as j −→ ∞) to the determinant of exact limits for particular subclasses of analytic functions on the unit disk U with specified values of j and m. When m = 2, j = 1, and b 1 = 1, the Hankel determinant is 2 | ≤ 1 is known (see [30]). Janteng et al [33] obtained sharp bounds for the functional H 2 (2) for the function f in the subclass RT of S, which was introduced by MacGregor [34] and consists of functions whose derivative has a positive real part.…”

Coefficient Bounds for Two Subclasses of Analytic Functions Involving a Limacon-Shaped Domain

“…It is constructive to identify whether certain coefficient functionals connected to the function f are bounded in the disc ∆ or not. In particular, Noor [3] investigated the asymptotic behavior as n → ∞ of H k,n ( f ). In [1], Pommerenke discusses a few of the applications of Hankel determinants for learning about the presence of singularities.…”

On q-Calculus Related Generalization of Close-to-Convexity

“…In this paper, we shall use the concept of q-calculus to define and study certain classes of analytic functions which are q-analogue of C, S , K and related generalizations. For the applications of q-calculus in geometric functions, see [11][12][13][14][15][16][17][18][19] and the references therein.…”