Let A be a positive bounded linear operator on a Hilbert space H,.,.\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( { {\mathscr {H}}},\left\langle .,.\right\rangle \right) $$\end{document}. The semi-inner product x,yA:=Ax,y\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left\langle x,y\right\rangle _{A}:=\left\langle Ax,y\right\rangle $$\end{document}, x, y∈\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\in $$\end{document}H\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${{\mathscr {H}}}$$\end{document}, induces a semi-norm .A\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left\| .\right\| _{A}$$\end{document} on H\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$${{\mathscr {H}}}$$\end{document}. Let ωA\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\omega _{A}$$\end{document}T\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( T\right) $$\end{document} denote the A -numerical radius of an operator T in semi-Hilbertian space H,.A\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\left( { {\mathscr {H}}},\left\| .\right\| _{A}\right) $$\end{document}. Our aim in this paper is to give new inequalities of A-numerical radius of operators in semi-Hilbertian spaces. In particular, we show that ωAnT≤12n-1ωATn+TA∑p=1n-112pωAn-p-1TTpA,\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$\begin{aligned} \omega _{A}^{n}\left( T\right) \le \frac{1}{2^{n-1}}\omega _{A}\left( T^{n}\right) +\left\| T\right\| _{A}\displaystyle \sum _{p=1}^{n-1}\frac{ 1}{2^{p}}\omega _{A}^{n-p-1}\left( T\right) \left\| T^{p}\right\| _{A}, \end{aligned}$$\end{document}for all n=2,3,…\documentclass[12pt]{minimal}
\usepackage{amsmath}
\usepackage{wasysym}
\usepackage{amsfonts}
\usepackage{amssymb}
\usepackage{amsbsy}
\usepackage{mathrsfs}
\usepackage{upgreek}
\setlength{\oddsidemargin}{-69pt}
\begin{document}$$n=2,3,\ldots $$\end{document} Further, an extension of some inequalities of bounded linear operators on a Hilbert space due to Dragomir (Inequalities for the numerical radius of linear operators in Hilbert spaces. Springer briefs in mathematics, Springer, Berlin, 2013; Tamkang J Math 39(1):1–7, 2008) and Kittaneh et al. (Linear Algebra Appl 471:46–53, 2015) are proved on a semi-Hilbert space and some more related results are also obtained.
