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Let $f$ be a transcendental meromorphic function, defined in the complex plane $mathbb{C}$. In this paper, we give a quantitative estimations of the characteristic function $T(r,f)$ in terms of the counting function of a homogeneous differential polynomial generated by $f$. Our result improves and generalizes some recent results.
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In this paper, we prove some value distribution results which lead to some normality criteria for a family of analytic functions. These results improve some recent results.
Let $f$ be a transcendental meromorphic function defined in the complex plane $mathbb{C}$. We consider the value distribution of the differential polynomial $f^{q_{0}}(f^{(k)})^{q_{k}}$, where $q_{0}(geq 2), q_{k}(geq 1)$ are $k(geq1)$ non-negative integers. We obtain a quantitative estimation of the characteristic function $T(r, f)$ in terms of $overline{N}left(r,frac{1}{f^{q_{_{0}}}(f^{(k)})^{q_{k}}-1}right)$.par Our result generalizes the results obtained by Xu et al. (Math. Inequal. Appl., 14, 93-100, 2011) and Karmakar and Sahoo (Results Math., 73, 2018) for a particular class of transcendental meromorphic functions.
Expressions for the summation of a new series involving the Laguerre polynomials are obtained in terms of generalized hypergeometric functions. These results provide alternative, and in some cases simpler, expressions to those recently obtained in the literature.
In this paper are discussed the results of new numerical experiments on zero distribution of type I Hermite-Pade polynomials of order $n=200$ for three different collections of three functions $[1,f_1,f_2]$. These results are obtained by the authors numerically and do not match any of the theoretical results that were proven so far. We consider three simple cases of multivalued analytic functions $f_1$ and $f_2$, with separated pairs of branch points belonging to the real line. In the first case both functions have two logarithmic branch points, in the second case they both have branch points of second order, and finally, in the third case they both have branch points of third order. All three cases may be considered as representative of the asymptotic theory of Hermite-Pade polynomials. In the first two cases the numerical zero distribution of type I Hermite-Pade polynomials are similar to each other, despite the different kind of branching. But neither the logarithmic case, nor the square root case can be explained from the asymptotic point of view of the theory of type I Hermite-Pade polynomials. The numerical results of the current paper might be considered as a challenge for the community of all experts on Hermite-Pade polynomials theory.
This is an auxiliary note to [12]. To be precise, here we have gathered the proofs of all the statements in [12, Section 5] that happen to have points of contact with techniques recently developed in Chousionis-Pratt [5] and Chunaev [6].
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