Techniques of the differential subordination for domains bounded by conic sections

Abstract

We solve the problem of finding the largest domain D for which, under given ψ and q, the differential subordination, where D and q(ᐁ) are regions bounded by conic sections, is satisfied. The shape of the domain D is described by the shape of q(ᐁ). Also, we find the best dominant of the differential subordination) maps the unit disk onto a conical domain contained in a right half-plane. Various applications in the theory of univalent functions are also given.

On λ-pseudo bi-starlike functions related to some conic domains

“…then the class S * ( φ) is associated with a cardioid [19]. We may consider also the functions φ k,α related to the conic sections, that were introduced and studied by Kanas et al [8] - [13], where (0 ≤ k < ∞, 0 ≤ α < 1) and where…”

Some Applications of Differential Subordination of p-Valent Functions

“…Several authors discussed the principle of differential subordination (e.g., [6], [16] and [17]) and others used this principle to solve some differential subordination problems (e.g., [7], [8], [9], [20], [21], [23] and [25] )…”

Inclusion relations for certain families of integral operators associated with conic regions

“…The classes k -CV and k -ST with k ≥ 0 denote the famous classes of k-uniformly convex and k-starlike functions, respectively, introduced by Kanas and Wisniowska, respectively. For some details see [4][5][6][7].…”