ON A COMPREHENSIVE CLASS OF ANALYTIC P-VALENT FUNCTIONS ASSOCIATED WITH SHELL-LIKE CURVE AND MODIFIED SIGMOID FUNCTION

In this paper, the authors introduce and study a new class of analytic p-valent functions and its connections with some famous subclasses of analytic and univalent functions associated with shell-like curve and modified sigmoid function in the open unit disk E= {z : z |<1}. In particular, the coefficient condition for function f(z) belonging to the class Bp (λ, β) is investigated using a succinct mathematical approach. In addition, as a special case, convex functions of order 1/4 are shown to be in the aforementioned class Bp(λ, β) in E. With the aid of subordination pri nciple, the authors obtain the first three Taylor-Maclaurin coefficients |ap+1 |, |ap+2 | and |ap+3 | as well as the Fekete-Szegö functional |ap+2 - η a2p+1| for functions f(z) belonging to the class Bp( λ, β, σ; p ̃ ) involving modified sigmoid function and associated with shell-like curve.

is said to be starlike, convex and bounded turning function of order β respectively, if the following geometric conditions are satisfied: Here, we denote the classes of starlike, convex and bounded turning functions of order β by ( ) ( A function ) (z f is said to be analytic p-valent in the open unit disk E, if it is analytic and assumes no value more than p-times for 1 < z .

Methodology
In this article, an analytic p-valent function is used to model two classes of analytic functions . Now for function f and g in E, there exists a function µ with the condition that Then we say that f is subordinate to g and it is denoted mathematically by Hamzat et al., Malaysian Journal of Computing, 7 (1): 995-1010, 2022 In particular, when g is univalent in E, then For recent studies on subordination, refer to (Hamzat & El-Ashwah, 2020 which are analytic such that (Duren, 1983). It is necessary to note that the correspondence between the class of Caratheodory functions Ρ and the class of Schwarz functions (functions with unit bound) , w exists and well known. That is, and 2 5 1 , 1 have been studied by different authors, see among others (Dzioket al., 2011a(Dzioket al., & 2011bOrhan et al., 2020;Raina & Sokol, 2016;Sokol, 1999 The theory of special function (logistic sigmoid function) has found its application in many physical problems such as in aerodynamics, thermodynamics and electrostatic potential to mention just a few. Logistic activation function is an information system that is inspired by the way nervous systems like the brain processes information. the most widely used sigmoid function is the logistic function which has the following series denotation see also Hamzat (2017) and Hamzat &Olayiwola (2017).In the present work, the following definitions shall be necessary.

Definition 2.1: The function
, of analytic p-valent functions, if the following geometric condition is satisfied:  (15), then the following class of analytic functions is obtained This class, ) , , is due to (Frasin & Jahangiri, 2009).
(c) If we set 0 = λ in (15), the following class of analytic functions is obtained (d) If we set 0 = λ and 1 = p in (15), then we obtain the following See Frasin & Darus (2001) and Nunokawa (1995) for more details on the class of analytic function defined in (d).

Main Results
In this section, sufficient conditions for functions ) (z f of the form (6)  However, before proceeding to the main results, the following Lemmas shall be necessary, the foremost is due to (Jack, 1971). See also Darus et al. (2015) Then ) (z ω is analytic in E and 0 ) 0 ( = ω . Using Logarithmic differentiation, we obtain Let there exists a point Then, it follows from Lemma 2.1 that Suppose that This contradicts the hypothesis of Lemma 2.2. Hence, we have for E z ∈ and this ends the proof.

Theorem 2.3: Suppose that
Then, ) (z q is analytic in E with 1 ) 0 ( = q . It follows from (22) and    Using Lemma 2.2 in (24), we can conclude that Therefore, we say that ) , ( β λ p B f ∈ and this completes the proof of Theorem 2.3. At this juncture, it is noteworthy to state some of the consequences of the above result.

Corollary 2.12: Suppose that
A

Corollary 2.14: Suppose that
. That is convex functions of order p.