In this paper, we obtain a result that improves the results of Govil and Nwaeze, Qazi and the classical result of Rivlin.

In the present article, we deal with the so-called overconvergence phenomenon in $\mathbb{C}$ of a slightly modified Post-Widder operator of real variable, that is with the extension of its approximation properties from the real axis in the complex plane. In this sense, error estimates in approximation and a quantitative Voronovskaya-type asymptotic formula are established.

Let $\mathcal{L} = −∆+V$ be a Schrödinger operator on $\mathbb{R}^n(n ≥ 3)$, where the nonnegative potential $V$ belongs to reverse Hölder class  $RH_{q_1}$ for $q_1 > \frac{n}{2}$. Let $H^p_{\mathcal{L}}(\mathbb{R}^n)$ be the Hardy space associated with $\mathcal{L}$. In this paper, we consider the commutator $[b,T_α]$, which associated with the Riesz transform $T_α = V^α(−∆+V)^{-\alpha}$ with $0<α≤ 1$, and a locally integrable function $b$ belongs to the new Campanato space $Λ^θ_β(ρ)$. We establish the boundedness of $[b,T_α]$ from $L^p(\mathbb{R}^n)$ to $L^q(\mathbb{R}^n)$ for $1<p<q_1/α$ with $1/q=1/p−β/n$. We also show that $[b,T_α]$ is bounded from $H^p_{\mathcal{L}}(R^n)$ to $L^q(\mathbb{R}^n)$ when $n/ (n+β) < p ≤ 1$, $1/q=1/p−β/n$. Moreover, we prove that $[b,T_α]$ maps $H^{\frac{n}{n+\beta}}_{\mathcal{L}}(\mathbb{R}^n)$ continuously into weak $L^1(\mathbb{R}^n)$.

In the present paper, we deal with Chlodowsky type generalization of the Baskakov operators, special case of these operators includes Chlodowsky type Meyer– König and Zeller operators (see [21]). With the help of Bohman-Korovkin theorem, we obtain some approximation properties for these operators. We give a modification of the operators in the space of differentiable functions and we also present examples of graphs for approximation. Finally, we apply these operators to the solution of the differential equation.

It is well-known that if $p$ is a homogeneous polynomial of degree $k$ in $n$ variables, $p∈\mathcal{P}_k$, then the ordinary derivative $p(∇) (r^{2−n})$ has the form $A_{n,k}\Upsilon(x)r^{2−n−2k}$ where $A_{n,k}$ is a constant and where $\Upsilon$ is a harmonic homogeneous polynomial of degree $k,$ $\Upsilon\in \mathcal{H}_k$, actually the projection of $p$ onto $\mathcal{H}_k$. Here we study the distributional derivative $p(\overline{∇})(r^{2−n})$ and show that the ordinary part is still a multiple of $\Upsilon$, but that the delta part is independent of $\Upsilon$, that is, it depends only on $p−\Upsilon$. We also show that the exponent $2−n$ is special in the sense that the corresponding results for $p(∇)(r^α)$ do not hold if $α\neq 2−n$.
Furthermore, we establish that harmonic polynomials appear as multiples of $r^{2−n−2k−2k'}$ when $p(∇)$ is applied to harmonic multipoles of the form $\Upsilon'(x)r^{2−n−2k'}$ for some $\Upsilon' \in \mathcal{H}_k$.

The purpose of this paper is to verify the Smulyan lemma for the support function, and also the Gateaux differentiability of the support function is studied on its domain. Moreover, we provide a characterization of Frechet differentiability of the support function on the extremal points.

In this paper, we have investigated second Hankel determinants and Fekete-Szegö inequalities for some subclasses of Bi-univalent functions with respect to symmetric and Conjugate points which are subordinate to a shell shaped region in the open unit disc ∆.

In this paper, using spectral decimation, we prove that the "hot spots" conjecture holds on a class of homogeneous hierarchical gaskets introduced by Hambly, i.e., every eigenfunction of the second-smallest eigenvalue of the Neumann Laplacian (introduced by Kigami) attains its maximum and minimum on the boundary.