• Shams, Saeid (Department of Mathematics University of Urmia) ;
  • Ebadian, Ali (Department of Mathematics Payame Noor University) ;
  • Sayadiazar, Mahta (Department of Mathematics University of Urmia) ;
  • Sokol, Janusz (Department of Mathematics Rzeszow University of Technology)
  • Received : 2013.01.18
  • Published : 2014.05.31


Let $\mathcal{A}$ be the class of analytic functions f in the open unit disk $\mathbb{U}$={z : ${\mid}z{\mid}$ < 1} with the normalization conditions $f(0)=f^{\prime}(0)-1=0$. If $f(z)=z+\sum_{n=2}^{\infty}a_nz^n$ and ${\delta}$ > 0 are given, then the $T_{\delta}$-neighborhood of the function f is defined as $$TN_{\delta}(f)\{g(z)=z+\sum_{n=2}^{\infty}b_nz^n{\in}\mathcal{A}:\sum_{n=2}^{\infty}T_n{\mid}a_n-b_n{\mid}{\leq}{\delta}\}$$, where $T=\{T_n\}_{n=2}^{\infty}$ is a sequence of positive numbers. In the present paper we investigate some problems concerning $T_{\delta}$-neighborhoods of function in various classes of analytic functions with $T=\{2^{-n}/n^2\}_{n=2}^{\infty}$. We also find bounds for $^{\delta}^*_T(A,B)$ defined by $$^{\delta}^*_T(A,B)=jnf\{{\delta}&gt;0:B{\subset}TN_{\delta}(f)\;for\;all\;f{\in}A\}$$ where A, B are given subsets of $\mathcal{A}$.


analytic functions;univalent;starlike;convex;close-to-convex;concave functions;neighborhood;$T_{\delta}$-neighborhood;T-factor


  1. J. Stankiewicz, Neighbourhoods of meromorphic functions and Hadamard products, Ann. Polon. Math. 46 (1985), 317-331.
  2. 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.
  3. T. Sheil-Small, On linear accessibility and the conformal mapping of convex domains, J. Analyse Math. 25 (1972), 259-276.
  4. T. Sheil-Small and E. M. Silvia, Neighborhoods of analytic functions, J. Analyse Math. 52 (1989), 210-240.
  5. L. Lewin, Dilogarithms and Associated Functions, Macdonald, London, 1958.
  6. St. Ruscheweyh, Neighborhoods of univalent functions, Proc. Amer. Math. Soc. 81 (1981), no. 4, 521-527.
  7. U. Bednarz, Stability of the Hadamard product of k-uniformly convex and k-starlike functions in certain neighbourhood, Demonstratio Math. 38 (2005), no. 4, 837-845.
  8. F. G. Avkhadiev, Ch. Pommerenke, and K. J. Wirths, On the coefficients of concave univalent functions, Math. Nachr. 271 (2004), 3-9.
  9. F. G. Avkhadiev, Ch. Pommerenke, and K. J. Wirths, Sharp inequalities for the coefficients of concave schlicht functions, Comment. Math. Helv. 81 (2006), no. 4, 801-807.
  10. F. G. Avkhadiev and K. J. Wirths, Convex holes produce lower bound for coefficients, Complex Var. Theory Appl. 47 (2002), no. 7, 553-563.
  11. 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.
  12. U. Bednarz and J. Sokol, On the integral convolution of certain classes of analytic functions, Taiwanese J. Math. 13 (2009), no. 5, 1387-1396.
  13. U. Bednarz and J. Sokol, T-neighborhoods of analytic functions, J. Math. Appl. 32 (2010), 25-32.
  14. 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.
  15. P. L. Duren, Univalent functions, Springer Verlag, Grund. math. Wiss. 259, New York, Berlin, Heidelberg, Tokyo, 1983.
  16. R. Fournier, A note on neighbourhoods of univalent functions, Proc. Amer. Math. Soc. 87 (1983), no. 1, 117-120.
  17. R. Fournier, On neighbourhoods of univalent starlike functions, Ann. Polon. Math. 47 (1986), no. 20, 189-202.
  18. R. Fournier, On neighbourhoods of univalent convex functions, Rocky Mountain J. Math. 16 (1986), no. 3, 579-589.

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