On the behaviour of analytic representation of the multivalent $$\alpha $$ α -convex functions

Vali Soltani Masih (), Jahangir Cheshmavar () and Saeideh Maghsoudi ()
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Vali Soltani Masih: Payame Noor University
Jahangir Cheshmavar: Payame Noor University
Saeideh Maghsoudi: Payame Noor University

Indian Journal of Pure and Applied Mathematics, 2025, vol. 56, issue 2, 659-675

Abstract: Abstract Let $$\mathscr {M}_p(\alpha ;\varphi )$$ M p ( α ; φ ) denote the class of p-valently $$\alpha $$ α -convex functions by $$h(\xi )=\xi ^p+\sum _{j=p+1}^{\infty }a_j\xi ^j$$ h ( ξ ) = ξ p + ∑ j = p + 1 ∞ a j ξ j in the open unit disk $$\textbf{D}$$ D , such that $$\begin{aligned} \frac{1}{p}\left( \left( 1-\alpha \right) \frac{\xi h'(\xi )}{h(\xi )}+\alpha \left( \frac{\xi h''(\xi )}{h'(\xi )}+1\right) \right) \prec \varphi (\xi )\ \left( \alpha \ge 0,\ p=1,2, \ldots \right) \end{aligned}$$ 1 p 1 - α ξ h ′ ( ξ ) h ( ξ ) + α ξ h ′ ′ ( ξ ) h ′ ( ξ ) + 1 ≺ φ ( ξ ) α ≥ 0 , p = 1 , 2 , … where, $$\varphi $$ φ is an analytic convex univalent function with positive real part and $$\varphi (0)=1$$ φ ( 0 ) = 1 . The symbol $$\prec $$ ≺ is the subordinate relation. In the present paper, subordination theorem, sharp bounds for the Fekete–Szegö coefficient functional associated to h and the integral representation of the functions to this class are given. Also, the monotony of class $$\mathscr {M}_p(\alpha ;\varphi )$$ M p ( α ; φ ) with respect to parameter $$\alpha $$ α is shown. Moreover, we determine the order of starlikeness for class $$\mathscr {M}_p(\alpha ;\varphi _{A,B})$$ M p ( α ; φ A , B ) , where, $$\begin{aligned} \varphi _{A,B}(\xi )=\frac{1+A \xi }{1+B \xi }, \ -1\le B

Keywords: $$p$$ p -Valently starlike and convex functions; p-Valently $$\alpha $$ α -convex function; Subordination; Extreme function; Hypergeometric functions; 30C45; 30C80 (search for similar items in EconPapers)
Date: 2025
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DOI: 10.1007/s13226-023-00509-9

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