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The well-known E -set introduced by Hayman in 1960 is a countable collection of Euclidean discs in the complex plane, whose subtending angles at the origin have a finite sum. An important special case of an E -set is known as the R-set, for which the sum of the diameters of the discs is finite. These sets appear in numerous papers in the theories of complex differential and functional equations. A given E -set (hence an R-set) has the property that the set of angles θ for which the ray arg (z) = θ meets infinitely many discs in the E -set has linear measure zero. This paper offers a continuous transition from E -sets to R-sets and then to much thinner sets. In addition to rays, plane curves that originate from the zero distribution theory of exponential polynomials will be considered. It turns out that almost every such curve meets at most finitely many discs in the collection in question. Analogous discussions are provided in the case of the unit disc D , where the curves tend to the boundary ∂ D tangentially or non-tangentially. Finally, these findings will be used for improving well-known estimates for logarithmic derivatives, logarithmic differences and logarithmic q-differences of meromorphic functions, as well as for improving standard results on exceptional sets. [ABSTRACT FROM AUTHOR]

The Hankel determinant H 2 , 1 F f - 1 / 2 of logarithmic coefficients is defined as H 2 , 1 F f - 1 / 2 : = Γ 1 Γ 2 Γ 2 Γ 3 = Γ 1 Γ 3 - Γ 2 2 , where Γ 1 , Γ 2 , and Γ 3 are the first, second, and third logarithmic coefficients of inverse functions belonging to the class S of normalized univalent functions. In this paper, we establish sharp inequalities H 2 , 1 F f - 1 / 2 ≤ 19 / 288 , H 2 , 1 F f - 1 / 2 ≤ 1 / 144 , and H 2 , 1 F f - 1 / 2 ≤ 1 / 36 for the logarithmic coefficients of inverse functions, considering starlike and convex functions, as well as functions with bounded turning of order 1/2, respectively. [ABSTRACT FROM AUTHOR]

In this paper, we first obtain a refined version of the Bohr inequality of norm-type for holomorphic mappings with lacunary series on the polydisk in $ \mathbb {C}^n $ C n under some restricted conditions. Next, we determine the refined version of the Bohr inequality for holomorphic functions defined on a balanced domain G of a complex Banach space X and take values in the unit disk $ \mathbb {D} $ D . Furthermore, as a consequence of one of these results, we obtain a refined version of the Bohr-type inequality for harmonic functions $ f=h+\bar {g} $ f = h + g ¯ defined on a balanced domain $ G\subset X $ G ⊂ X. All the results are proved to be sharp. [ABSTRACT FROM AUTHOR]

In this paper, we consider the class of polynomials P (z) : = ∑ j = 0 n c j z j having all zeros in a closed disk | z | ≤ k , where k ≥ 1 and obtain a result that improves and generalizes the results of Govil, Jain and others by using certain coefficients of P(z). [ABSTRACT FROM AUTHOR]

If $ P(z)=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a n ∏ v = 1 n (z − z v) is a polynomial of degree n having all its zeros in $ |z|\le k, k\ge 1 $ | z | ≤ k , k ≥ 1 then Aziz [Inequalities for the derivative of a polynomial. Proc Am Math Soc. 1983;89(2):259–266] proved that \[ \max_{|z|=1}|P'(z)|\ge \frac{2}{1+k^n}\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \] max | z | = 1 | P ′ (z) | ≥ 2 1 + k n ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. Recently, Kumar [On the inequalities concerning polynomials. Complex Anal Oper Theory. 2020;14(6):1–11 (Article ID 65)] established a generalization of this inequality and proved under the same hypothesis for a polynomial $ P(z)=a_0+a_1z+a_2z^2+\cdots +a_nz^n=a_n\prod _{v=1}^{n}(z-z_v) $ P (z) = a 0 + a 1 z + a 2 z 2 + ⋯ + a n z n = a n ∏ v = 1 n (z − z v) , that $$\begin{align*} & \max_{|z|=1}|P'(z)| \\ & \ge \left(\frac{2}{1+k^n}+\frac{(|a_n|k^n-|a_0|)(k-1)}{(1+k^n)(|a_n|k^n+k|a_0|)}\right)\sum_{v=1}^{n}\frac{k}{k+|z_v|}\max_{|z|=1}|P(z)|. \end{align*} $$ max | z | = 1 | P ′ (z) | ≥ (2 1 + k n + (| a n | k n − | a 0 |) (k − 1) (1 + k n) (| a n | k n + k | a 0 |)) ∑ v = 1 n k k + | z v | max | z | = 1 | P (z) |. In this paper, we sharpen the above inequalities and further extend the obtained results to the polar derivative of a polynomial. As a consequence, our results also sharpens considerably some results of Dewan and Upadhye [Inequalities for the polar derivative of a polynomial. J Ineq Pure Appl Math. 2008;9:1–9 (Article ID 119)]. [ABSTRACT FROM AUTHOR]

In this paper, we study the behavior of derivatives of algebraic polynomials in bounded and unbounded regions of the complex plane. At the same time, both interior and exterior cusp points are allowed on the boundary of such regions. Bernstein-Walsh-type estimates are obtained for derivatives of algebraic polynomials in the specified region with corners for exterior points, as well as Markoff-type estimates for closure of the region. As a result, estimates are found for the derivatives of algebraic polynomials in the whole complex plane. It is also shown that the inequality for the closed region is exact in order for the given region. [ABSTRACT FROM AUTHOR]

The main aim of this paper is to prove a generalization of the classical Bohr theorem and as an application, we obtain a counterpart of Bohr theorem for the generalized Cesáro operator. [ABSTRACT FROM AUTHOR]

In this paper, we prove some inequalities that relate the sup-norm of a univariate complex coefficient polynomial and its derivative, when there is a restriction on its zeros. We further obtain the polar derivative generalizations of the obtained results. The obtained results produce various inequalities that are sharper than the previous ones while taking into account the placement (absolute value) of the zeros and the extremal coefficients of the polynomial. Moreover, some concrete numerical examples are presented, showing that in some situations, the bounds obtained by our results can be considerably sharper than the ones previously known. [ABSTRACT FROM AUTHOR]

In this paper, we first determine Bohr’s inequality for the class of harmonic mappings f = h + g ¯ in the unit disk D , where either both h (z) = ∑ n = 0 ∞ a p n + m z p n + m and g (z) = ∑ n = 0 ∞ b p n + m z p n + m are analytic and bounded in D , or satisfies the condition | g ′ (z) | ≤ d | h ′ (z) | in D \ { 0 } for some d ∈ [ 0 , 1 ] and h is bounded. In particular, we obtain Bohr’s inequality for the class of harmonic p-symmetric mappings. Also, we investigate the Bohr-type inequalities of harmonic mappings with a multiple zero at the origin and that most of results are proved to be sharp. [ABSTRACT FROM AUTHOR]