Some Marx-Strohhacker type results for a class of multivalent functions

Abstract

In this paper, two Marx-Strohhäcker type results are proven for a certain class of analytic and multivalent functions in the open unit disk D. The first result gives the order of multivalent starlikeness for multivalently convex functions of some specified order. The second result provides a lower bound over the unit disk D of <  f .´/ p à for functions f .´/ that are multivalently starlike of a given order. Relevant connections of the results presented here with earlier results are also indicated.

Marx-Strohhäcker theorem for Multivalent Functions

“…Although the original proof of theorem was complicated, Miller and Mocanu [5, Section 2.6, p. 56] gave simple algebraic proof using the technique of differential subordination. In 2017, Nunokawa et al [8] extended some of these results for multivalent functions. In this paper, we extend some other forms of Marx-Strohhäcker type results for multivalent functions.…”

“…Nunokawa et al [8] extended these differential implications for multivalent functions f ∈ A p (p ≥ 2) by finding β and γ such that…”

“…for all z ∈ D. More details of p-valent starlike and p-valent convex functions can be found in [7][8][9]12]. For f ∈ A, Marx-Strohhäcker theorem asserts that, for all z ∈ D,…”

An application on differential equations of order m

“…Therefore, various subclasses of p-valent functions in U was studied by Altıntaş et al in [8], Nunokawa et al in [12] and Srivastava et al in [16,17].…”

Certain family of integral operators associated with multivalent functions preserving subordination and superordination

“…If Ψ(z) is not subordinate to Φ(z), by using of the result of [13, p. 24] (see also [16]), we know that there exist two points z 0 ∈ U and ζ 0 ∈ ∂U such that…”