Applications of Fractional Differential Operator to Subclasses of Uniformly q-Starlike Functions

Abstract

In this paper, we use the concept of quantum (or q-) calculus and define a q-analogous of a fractional differential operator and discuss some of its applications. We consider this operator to define new subclasses of uniformly q-starlike and q-convex functions associated with a new generalized conic domain, Λβ,q,γ. To begin establishing our key conclusions, we explore several novel lemmas. Furthermore, we employ these lemmas to explore some important features of these two classes, for example, inclusion relations, coefficient bounds, Fekete–Szego problem, and subordination results. We also highlight many known and brand-new specific corollaries of our findings.

Results of Third-Order Strong Differential Subordinations

“…In Theorem 1, inequalities ( 17) and ( 18) are the most necessary for solving thirdorder differential subordination. If third-order terms in (18) are missing, then they are not required to satisfy (17).…”

“…In this particular case, Theorem 4 becomes Theorem 5. Consider that the q given in ( 14) and p ∈ H[b, n] satisfy (17).…”

“…The Sȃlȃgean and Ruscheweyh operators were used together in the study presented in [11], and a multiplier transformation provided new strong differential subordinations in [12][13][14]. The Komatu integral operator was applied for obtaining new strong differential subordinations results [15,16], and other differential operators proved effective for studying strong differential subordinations [17]. The fractional derivative operator was used in [18], and the fractional integral of the extended Dziok-Srivastava operator was used in [19].…”

New Applications of Fractional q-Calculus Operator for a New Subclass of q-Starlike Functions Related with the Cardioid Domain

“…Fractional differential calculus now plays a significant part in the diagnosis of diseases in the medical field, (see [57][58][59][60]). For the area of complex fractional differential operators, see the following articles [61][62][63].…”

Applications of Horadam Polynomials for Bazilevič and λ-Pseudo-Starlike Bi-Univalent Functions Associated with Sakaguchi Type Functions

“…This functional has garnered significant attention, especially in the study of various subfamilies of univalent functions. Geometric function theory researchers are now actively exploring this subject [16][17][18][19][20][21][22][23].…”