Neighbourhoods of meromorphic functions and Hadamard products

On neighborhoods of analytic functions with positive real part

“…If T = { n } ∞ n = 2 then the TN δ ‐neighborhood becomes the δ‐neighborhood N δ ( f ) introduced in 6, 7 by St. Ruscheweyh who used it to generalize the earlier result that \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$N_1(z)\subseteq \cal {S^*}\!$\end{document}, obtained in 4, by showing that if \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$f\in \cal {C}$\end{document} then \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$N_{1/4}(f)\subseteq \cal {S^*}$\end{document}, where \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\cal {C}$\end{document}, \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\cal {S^* }$\end{document} denote the well‐known classes of convex and starlike functions, respectively. Some results of this type one can find in 1–3, 10. The TN δ ‐neighborhood was introduced in 8, where the authors considered the problem of finding a sufficient condition on \documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$f\in {\cal A}$\end{document} that implies the existence of TN δ ( f ) being contained in a given subclass.…”

T-Neighborhoods in Various Classes of Analytic Functions

“…St. Ruscheweyh in [14] considered T = {n} ∞ n=2 and showed that if f ∈ C, then T N 1/4 (f ) ⊂ S * , where C, S * denote the well known classes of convex and starlike functions, respectively. In [4,5,6,7,10,11,12,17,18] other authors investigated some interesting results concerning neighborhoods of several classes of analytic functions. Some of the relations between the neighborhoods for a certain class of analytic functions was described by S. Shams et al [15].…”

Stability of the Integral Convolution of k-Uniformly Convex and k-Starlike Functions

“…Problem of neighbourhoods in various classes of functions was also studied in papers [1], [7], [10], [15], [16].…”

“…eg. [13], [15]). Let A * B ⊂ C, the convolution (1.4) is called C-stable on the pair of classes (A, B) if there exists δ > 0 such that N δ (A) * N δ (B) ⊂ C and C-unstable otherwise (cf.…”

“…Stability of inclusions for integral convolution is defined in a similar way. The constant δ which characterizes the stability of Hadamard or integral convolutions is defined as 15) respectively. If the related value (1.14) or (1.15) is positive then there exists a neighborhood of the class A and B mapped by the convolution or the integral convolution into C. The stability of the Hadamard or integral convolution can be regarded as a problem of preserving a product of the topology of neighbourhoods in A and B onto the product A * B, and A ⊗ B, respectively.…”