Abstract

In this paper, we give some Young type inequalities for scalars. By using these inequalities we establish corresponding Young type inequalities for matrices. In addition, we present some reverses of the Young type inequalities and give several refinements for matrices.

Keywords: Heinz mean inequalities; positive semidefinite matrices; Hilbert-Schmidt norm; Young inequality.

MSC: 26D07, 26D15, 15A18, 47A63

Abstract

In this work, we study an inertial extragradient-like S-iteration process for approximating a common element of the set of solutions of some variational inequality problem involving a monotone Lipschitz map and a fixed point of asymptotically nonexpansive mapping in a reflexive Banach space. The result in this paper is an extension and generalization of some recently announced results.

Keywords: Strong convergence; fixed point problem; variational inequality; Bregman quasi nonexpansive mappings.

MSC: 47H09, 47H10, 47J25

Abstract

We study the linear combinations $f(z)= \lambda f_{1}(z)+(1-\lambda) f_{2}(z)$ of two univalent harmonic mappings $f_{1}$ and $f_{2}$ in the cases when $\lambda$ is some complex number. We determine the radius of close-to-convexity of $f$ and establish some sufficient conditions for $f$ to be locally-univalent and sense-preserving. Some known results reduce to particular cases of our general results.

Keywords: Univalent harmonic mappings; linear combination; convex in the horizontal direction.

MSC: 30C45

Abstract

In the present paper, we investigate geometric properties of Clairaut anti-invariant submersions whose total spaces are nearly Kähler manifolds. We obtain a condition for a Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers.

Keywords: Riemannian submersion; nearly Kähler manifolds; anti-invariant submersion; Clairaut submersion; totally geodesic maps.

MSC: 53C12, 53C15, 53C20, 53C55

Abstract

Let $P(z)=\sum_{\nu=0}^n a_\nu z^\nu$ be a polynomial of degree $n$ having all its zeros in $|z|\leq k$, $ k\geq 1$. It was shown by Govil that $\underset{|z|=1}\max|P'(z)|\geq\frac{n}{1+k^n}\underset{|z|=1}\max|P(z)|$. In this paper, we shall obtain some sharp estimates by involving the coefficients which not only refine the above result but also generalise some well-known results of this type.

Keywords: Polynomials; inequalities in complex domain; derivative; s-fold zeros.

MSC: 26D10, 30C15, 41A17

Abstract

One of the initial and at the same time key formulas of integral geometry is Crofton's formula. We consider a complex version of Crofton's formula.

Keywords: Integral geometry; Crofton formula; complex analysis; Fubini–Study metric.

MSC: 53C65, 32F45

Abstract

In this article, we introduce the notion of $\mathcal{I}$-statistical convergence of sequences as one of the extensions of $\mathcal{I}$-convergence in the gradual normed linear spaces. We investigate some fundamental properties of the newly introduced notion and its relation with some other methods of convergence. Also we introduce and investigate the concept of $\mathcal{I}$-statistical limit points, cluster points and establish some implication relations.

Keywords: Gradual number; gradual normed linear spaces; statistical convergence; $\mathcal{I}$-convergence; $\mathcal{I}$-statistical convergence.

MSC: 03E72, 40A35, 40A05