New inequalities in semi-Hilbert spaces with joint numerical radius applications

Let A be a nonzero positive operator on a complex Hilbert space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>,</m:mo> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">〈</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo>,</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">〉</m:mo> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(H,\,·\,,·\,)} , inducing the semi-inner product <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">〉</m:mo> </m:mrow> <m:mi>A</m:mi> </m:msub> <m:mo>:=</m:mo> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mrow> <m:mi>A</m:mi> <m:mo>⁢</m:mo> <m:mi>x</m:mi> </m:mrow> <m:mo>,</m:mo> <m:mi>y</m:mi> <m:mo stretchy="false">〉</m:mo> </m:mrow> </m:mrow> </m:math> { x,yA:= Ax,y} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>y</m:mi> </m:mrow> <m:mo>∈</m:mo> <m:mi mathvariant="script">ℋ</m:mi> </m:mrow> </m:math> {x,y} . The space <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mi mathvariant="script">ℋ</m:mi> <m:mo>,</m:mo> <m:mo>∥</m:mo> <m:mo>⋅</m:mo> <m:msub> <m:mo>∥</m:mo> <m:mi>A</m:mi> </m:msub> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> {(H,\|·\|A)} , where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mrow> <m:mo>∥</m:mo> <m:mi>x</m:mi> <m:mo>∥</m:mo> </m:mrow> <m:mi>A</m:mi> </m:msub> <m:mo>=</m:mo> <m:msqrt> <m:msub> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:mi>x</m:mi> <m:mo stretchy="false">〉</m:mo> </m:mrow> <m:mi>A</m:mi> </m:msub> </m:msqrt> </m:mrow> </m:math> {\|x\|A=√{ x,xA}} , is a semi-Hilbert space equipped with a seminorm induced by <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mrow> <m:mo rspace="4.2pt" stretchy="false">〈</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo>,</m:mo> <m:mo rspace="4.2pt">⋅</m:mo> <m:mo stretchy="false">〉</m:mo> </m:mrow> <m:mi>A</m:mi> </m:msub> </m:math> {\,·\,,·\,A} . In this paper, we establish the inequality <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mrow> <m:msup> <m:mrow> <m:mo maxsize="260%" minsize="260%">|</m:mo> <m:mrow> <m:munderover> <m:mo largeop="true" movablelimits="false" symmetric="true">∑</m:mo> <m:mrow> <m:mi>i</m:mi> <m:mo>=</m:mo> <m:mn>1</m:mn> </m:mrow> <m:mi>n</m:mi> </m:munderover> <m:mrow> <m:msub> <m:mi>c</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo>⁢</m:mo> <m:msub> <m:mrow> <m:mo stretchy="false">〈</m:mo> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:msub> <m:mi>y</m:mi> <m:mi>i</m:mi> </m:msub> <m:mo stretchy="false">〉</m:mo> </m:mrow> <m:mi>