In this paper, we introduce and investigate a new subclass of the function class $\Sigma$ of bi-univalent functions of the Mocanu-convex type defined in the open unit disk, satisfy Ma and Minda subordination conditions. Furthermore, we find estimates on the Taylor-Maclaurin coefficients $|a_2|$ and $|a_3|$ for functions in the new subclass introduced here. Further Application of Hohlov operator to this class is obtained. Several (known or new) consequences of the results are also pointed out.

In this paper, we will use a method of sharp maximal function approach to show the boundedness of commutator $[b,T_{R}^{\delta}]$ by Bochner-Riesz operators and the function $b$ on weighted Morrey spaces $L^{p,\kappa}(\omega)$ under appropriate conditions on the weight $\omega$,  where $b$ belongs to Lipschitz space or weighted Lipschitz space.

We analyze the local behavior of the Hausdorff centered measure for self-similar sets. If $E$ is a self-similar set satisfying the open set condition, then$$C^s(E \cap B(x,r)) \le (2r)^s$$for all $x \in E$ and $r >0$, where $C^s$ denotes the $s$-dimensional Hausdorff centered measure. The above inequality is used to obtain the upper bound of the Hausdorff centered measure. As the applications of above inequality, We obtained the upper bound of the Hausdorff centered measure for some self-similar sets with Hausdorff dimension equal to 1, and prove that the upper bound reach the exact Hausdorff centered measure.

In this paper we study the problem of explicit representation and convergence of Pál type (0;1) interpolation and its converse, with some additional conditions, on the non-uniformly distributed nodes on the unit circle obtained by projecting the interlaced zeros of $P_n(x)$ and $P′_n(x)$ on the unit circle. The motivation to this problem can be traced to the recent studies on the regularity of Birkhoff interpolation and Pál type interpolations on non-uniformly distributed zeros on the unit circle.

Using product and convolution theorems on Lorentz spaces, we characterize the sufficient and necessary conditions which ensure the validity of the doubly weighted Hardy-Littlewood-Sobolev inequality. It should be pointed out that we consider whole ranges of $p$ and $q$, i.e., $0< p\le \infty$ and $0< q\le \infty$.

This paper is devoted to studying direct and converse approximation theorems of the generalized Bernstein operators $C_{n}(f,s_{n},x)$ via so-called unified modulus$\omega_{\varphi^{\lambda}}^{2}(f,t)$, $0\leq\lambda\leq1$. We obtain  main results as follows$$ \omega_{\varphi^{\lambda}}^{2}(f,t)=O(t^{\alpha})\Longleftrightarrow|C_{n}(f,s_{n},x)-f(x)|=\mathcal{O}\big((n^{-\frac{1}{2}}\delta_{n}^{1-\lambda}(x))^{\alpha}\big),$$where $\delta_{n}^{2}(x)=\max\{\varphi^{2}(x),{1}/{n}\}$ and $0<\alpha<2$.

In this paper, we introduce the class of $n$-normed generalized difference sequences related to $\ell_p$-space. Some properties of this sequence space like solidness, symmetricity, convergence-free etc. are studied. We obtain some inclusion relations involving this sequence space.

In this paper, we will prove the boundedness of Hardy type operators $H_{\beta(x)}$ and $H^{\ast}_{\beta(x)}$of variable order $\beta(x)$ on Herz spaces $K^{\alpha(\cdot)}_{p(\cdot), q}$and $\dot{K}^{\alpha(\cdot)}_{p(\cdot), q}$, where $\alpha(\cdot)$ and $p(\cdot)$ are both variable.

We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide "computable" estimates on the operator norm.