The main focus of the present paper is to establish sufficient conditions for the parameters of the normalized form of the generalized Le Roy-type Mittag-Leffler function have certain geometric properties like close-to-convexity, univalency, convexity and starlikeness inside the unit disc. The results are new and their usefulness is depicted by deducing several interesting corollaries. The results improve several results available in the literature for the Mittag-Leffler function.
Banach [1] proved that good differential properties of function do not guarantee the a.e. convergence of the Fourier series of this function with respect to general orthonormal systems (ONS). On the other hand it is very well known that a sufficient condition for the a.e. convergence of an orthonormal series is given by the Menshov–Rademacher Theorem. The paper deals with sequence of positive numbers ((d_{n})) such that multiplying the Fourier coefficients ((C_{n}(f))) of functions with bounded variation by these numbers one obtains a.e. convergent series of the form (sum_{n=1}^{infty}d_{n}C_{n}(f)varphi_{n}(x).) It is established that the resulting conditions are best possible.
Let (mathcal{L}=-Delta+V) be the Schrödinger operator on (mathbb{R}^{n},) where (ngeq 3,) and nonnegative potential (V) belongs to the reverse Hölder class (RH_{q}) with (n/2leq q<n.) Let (H^{p}_{mathcal{L}}(mathbb{R}^{n})) denote the Hardy space related to (mathcal{L}) and (BMO_{mathcal{L}}(mathbb{R}^{n})) denote the dual space of (H^{1}_{mathcal{L}}(mathbb{R}^{n}).) In this paper, we show that (T_{alpha,beta}=V^{alpha}nablamathcal{L}^{-beta}) is bounded from (H^{p_{1}}_{mathcal{L}}(mathbb{R}^{n})) into (L^{p_{2}}(mathbb{R}^{n})) for (dfrac{n}{n+delta^{prime}}<p_{1}leq 1) and (dfrac{1}{p_{2}}=dfrac{1}{p_{1}}-dfrac{2(beta-alpha)}{n},) where (delta^{prime}=min{1,2-n/q_{0}}) and (q_{0}) is the reverse Hölder index of (V.) Moreover, we prove (T^{*}_{alpha,beta}) is bounded on (BMO_{mathcal{L}}(mathbb{R}^{n})) when (beta-alpha=1/2.)
In this paper, we investigate the uniqueness of transcendental meromorphic functions sharing three values with their derivatives in an arbitrary small angular domain including a Borel direction. The results extend the corresponding results from Gundersen, Mues and Steinmetz, Zheng, Li et al., and Chen.
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