On Starlike Functions of Negative Order Defined by q-Fractional Derivative

Abstract

In this paper, two new classes of q-starlike functions in an open unit disc are defined and studied by using the q-fractional derivative. The class Sq*˜(α), α∈(−3,1], q∈(0,1) generalizes the class Sq* of q-starlike functions and the class Tq*˜(α), α∈[−1,1], q∈(0,1) comprises the q-starlike univalent functions with negative coefficients. Some basic properties and the behavior of the functions in these classes are examined. The order of starlikeness in the class of convex function is investigated. It provides some interesting connections of newly defined classes with known classes. The mapping property of these classes under the family of q-Bernardi integral operator and its radius of univalence are studied. Additionally, certain coefficient inequalities, the radius of q-convexity, growth and distortion theorem, the covering theorem and some applications of fractional q-calculus for these new classes are investigated, and some interesting special cases are also included.

Hankel Determinants and Coefficient Estimates for Starlike Functions Related to Symmetric Booth Lemniscate

“…From the above discussion, it can be also known that t > 548 299 and when ũ → 0, then t → 1.433783077. Thus, we conclude that a possible solution exists in [1.433783077, 2) × 0, 7 18 , for inequality (23). A computation shows ∂G ∂t | y=y 0 = 0, in this interval.…”

“…Since g (u) < 0 for (0, 1), g(u) is decreasing in (0, 1), and so t 2 > 7 4 . A simple exercise shows that (23) does not hold in this case for all values of u ∈ 7 18 , 1 ; thus, there are no critical points of G in (0, 2) × 7 18 , 1 × (0, 1). Suppose that there is a critical point ( t, ũ, ṽ) of G existing in the interior of cuboid S. Clearly, it must satisfy that ũ < 7 18 .…”

“…Furthermore, factional and q-fractional derivatives are applied to the classes defined by using the above domains to study various generalizations of the classes of univalent functions. For some details on the applications of fractional operators, we refer to [19][20][21][22][23][24].…”

Coefficient Inequalities of q-Bi-Univalent Mappings Associated with q-Hyperbolic Tangent Function

“…Remarkably, as q approaches 1, the D q reduces to the classical derivative. For more details and recent applications of the q-fractional derivative, we refer the readers to [15][16][17][18][19][20][21] and the references therein.…”

Multivalent Functions and Differential Operator Extended by the Quantum Calculus

“…This calculus proved its efficiency and accuracy to generalize the families of differential and integral operators in a complex domain. In addition, special functions (see [7,8]) have associated with this calculus, especially the queen of special functions: Mittag-Leffler function (see [9][10][11][12]). The quantum calculus (q-calculus) has tremendous applications in different fields, for example, integral inequalities [13], summability [14], approximation and polynomials [15], and sequence spaces [16].…”