Generalization properties of Bernardi integral operator

Year 2025, Volume: 74 Issue: 3, 364 - 374, 23.09.2025

Muhammet Kamali , Hatun Özlem Guney , Shigeyoshi Owa

https://doi.org/10.31801/cfsuasmas.1521845

Abstract

Let $A\left( n\right)$ be the class of analytic functions $f\left(z\right)$ of the form
\begin{equation*}
f\left( z\right) =z+\sum_{k=n}^{\infty }a_{k}z^{k},\;\;\left(
n=2,3,4,...\right)
\end{equation*}
in the open unit disk $U.$ We introduce the integral operator $B_{j}f\left(
z\right) $$=B\left( B_{j-1}f\left( z\right) \right) $, $B_{1}f\left(
z\right) =Bf\left( z\right)$ and $B_{0}f\left( z\right) =f\left( z\right) $. In the present paper, we define the subclass $M_{j}\left( n,\gamma ,\alpha
\right) $ and discuss some interesting properties of $f\left( z\right) \in
A\left( n\right)$ concerning with the class $M_{j}\left( n,\gamma ,\alpha
\right) .$

Keywords

Analytic function , Bernardi integral operator , coefficient problem , argument problem , subordination

References

• Bernardi, S. D., Convex and starlike univalent functions, Trans. Amer. Math. Soc., 135 (1969), 429-446.
• Caratheodory, C., Uber der Variabilitatsbereich der Fourier’schem Konstanten won positiven harmonischen Funktionen, Rend. Circ. Mat. Palermo, 32 (1911), 193-217.
• Fejer, L., Riesz, F., Über einige funktionentheoretische Ungleichungen, Math. Z., 11 (1921), 305-314.
• Gwynme, E., The Poisson integral formula and representation of SU(1,1), Rose-Hulman Undergraduate Math. J., 12 (2011), 1-20.
• Hallenbeck, D. J., Ruscheweyh S., Subordination by convex functions, Proc. Amer. Math. Soc., 52 (1975), 191-195.
• Miller, S. S., Mocanu, P. T., Differential Subordinations, Theory and Applications, Series on Monographs and Textbook in Pure and Applied Mathematics, No. 225, Marcel Dekker, New York and Basel, 2000.
• Suffridge, T. J., Some remarks on convex maps of the unit disk, Duke Math. J., 37 (1970), 775-777.
• Tsuji, M., Complex Function Theory, Tokyo, Japanese, Maki Book Comp., 1968.