A generalization of the Bohr-Rogosinski sum

Abstract

In this paper, we investigate the Bohr-Rogosinski sum and the classical Bohr sum for analytic functions defined on the unit disk in a general setting. In addition, we discuss a generalization of the Bohr-Rogosinski sum for a class of analytic functions subordinate to the univalent functions in the unit disk. Several well-known results are observed from the consequences of our main results.

Bohr-Rogosinski type inequalities for concave univalent functions

“…The number r = 1/2 is called the Rogosinki Radius for the family B. The Bohr-Rogosinki inequality, which is considered by Kayumov et al in [27], is given by 2) has been studied by Kumar and Sahoo [35], and Liu et al [37].…”

A generalized Bohr-Rogosinski phenomenon

“…Recently, Kumar and Sahoo [29] obtained the generalized classical Bohr's Theorem for functions satisfying ℜ(f )(z) ≤ 1. Also see, Kayumov et al [28].…”

“…In context of the above problem and Muhanna [19], we now describe the notion of generalized Bohr-Rogosinski phenomenon here below, in terms of subordination, following the recent development as seen in [28,29,30]. Then we say S(f ) satisfies the Generalized Bohr-Rogosinski phenomenon, if there exists a constant r 0 ∈ (0, 1] such that…”

“…[29, Theorem 2.2] Let {ν k (r)} ∞ k=0 be a sequence of nonnegative continuous functions in [0, 1) such that the seriesν 0 (r) + ∞ k=1 ν k (r)converges locally uniformly with respect to r ∈ [0, 1). Let f (z) = ∞ n=0 a n z n with ℜf (z) ≤ 1 and p ∈ (0, 1].…”

A generalization of the Bohr inequality for bounded analytic functions on simply connected domains and its applications

“…For background information about this inequality and further work related to Bohr's radius, we refer survey article [5] and references therein. Moreover, to find certain recent results, we refer to [3,4,12,22,26,28,30,31]. Many interesting extensions of Bohr's inequality in various settings have been developed by several mathematicians.…”