On T-neighborhoods of various classes of analytic functions

Ghaderi, Nassireh

doi:10.22075/ijnaa.2025.35301.5270

On T-neighborhoods of various classes of analytic functions

Articles in Press

Document Type : Research Paper

Author

Nassireh Ghaderi

Department of Mathematics, Faculty of Science, Farhangian University, Sanandaj, Iran

10.22075/ijnaa.2025.35301.5270

Abstract

Let $\mathcal A$ be the class of analytic functions $f$ in the open unit disk ${\mathbb U}=\{z:\ |z|<1\}$ with the normalization conditions $f(0)=0$, $f'(0)=1.$ If $f(z) = z+ \sum_{n=2}^{\infty} a_n z^n$ and $\delta > 0$ are given, then the $T_\delta$-neighborhood of the function $f $ is defined as
$$TN_\delta (f) = \left\{g(z)= z+ \sum_{n=2}^{\infty} b_n z^n \in \mathcal{ A} : \sum_{n=2}^{\infty} T_n\left|a_n - b_n\right| \leq \delta \right\},$$
where $T= \left\{T_n\right\}_{n=2}^{\infty}$ is a sequence of positive numbers. In the present paper we investigate some problems concerning $T_{\delta}-$neighborhoods of analytic functions with $T = \left\{\frac{n^2}{3^n n!}\right\}_{n=2}^{\infty}$. One of the considered problems is to find a number $\delta^\ast_T(A, B)$ such that $$\delta^\ast_T(A, B)= \inf \left\{\delta >0 : B \subset TN_\delta(f) \: \textnormal{for all} \: f \in \mathcal A\right\},$$ where the sets $A,B \in \mathcal A$ are given.

Keywords

References

[1] F.G. Avkhadiev, Ch. Pommerenke, and K.J. Wirths, On the coefficients of concave univalent functions, Math. Nachr. 271 (2004), 3–9.

[2] F.G. Avkhadiev, Ch. Pommerenke and K.-J. Wirths, Sharp inequalities for the coefficients of concave Schlicht functions, Comment. Math. Helv. 81 (2006), 801–807.

[3] F.G. Avkhadiev and K.J. Wirths, Convex holes produce lower bound for coefficients, Complex Var. Theory Appl. 47 (2002), no. 7, 553–563.

[4] U. Bednarz, Stability of the Hadamard product of k-uniformly convex and k-starlike functions in certain neighbourhood, Demonstr. Math. 38 (2005), no. 4, 837–845.

[5] U. Bednarz and S. Kanas, Stability of the integral convolution of k-uniformly convex and k-starlike functions, J. Appl. Anal. 10 (2004), no. 1, 105–115.

[6] U. Bednarz and J. Sokól, On the integral convolution of certain classes of analytic functions, Taiwanese J. Math. 13 (2009), no. 5, 1387–1396.

[7] U. Bednarz and J. Sokol, T-neighborhoods of analytic functions, J. Math. Appl. 32 (2010), 25–32.

[8] A. Bielecki and Z. Lewandowski, Sur une generalisation de quelques theoremes de M. Biernacki sur les fonctions analytiques, Ann. Polon. Math. 12 (1962), 65–70.

[9] P.L. Duren, Univalent Functions, Springer Verlag, Grund. Math. Wiss. 259, New York, Berlin, Heidelberg, Tokyo, 1983.

[10] R. Fournier, A note on neighbourhoods of univalent functions, Proc. Amer. Math. Soc. 87 (1983), no. 1, 117–120.

[11] R. Fournier, On neighbourhoods of univalent starlike functions, Ann. Polon. Math. 47 (1986), no. 20, 189–202.

[12] R. Fournier, On neighbourhoods of univalent convex functions, Rocky Mountain J. Math. 16 (1986), no. 3, 579–589.

[13] L. Lewin, Dilogarithms and Associated Functions, Macdonald, London, 1958.

[14] St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 521–527.

[15] S. Shams and S.R. Kulkarni, Certain properties of the class of univalent functions defined by Ruscheweyh derivative, Bull. Cal. Math. Soc. 97 (2005), no. 3, 223–234.

[16] S. Shams, A. Ebadian, M. Sayadiazar, and J. Sokol, T-neighborhoods in various classes of analytic functions, Bull. Korean Math. Soc. 51 (2014), no. 3, 659–666.

[17] T. Sheil-Small, On linear accessibility and the conformal mapping of convex domains, J. Anal. Math. 25 (1972), 259–276.

[18] T. Sheil-Small and E. M. Silvia, Neighborhoods of analytic functions, J. Anal. Math. 52 (1989), 210–240.

[19] J. Stankiewicz, Neighbourhoods of meromorphic functions and Hadamard products, Ann. Polon. Math. 46 (1985), 317–331