Numerical radius and norm inequalities via a Cauchy-Schwarz extension

The generalization of inequalities involving the numerical radius is a well-established topic in functional analysis. In this article, we present refinements to certain known generalizations by employing an extension of the Cauchy–Schwarz inequality formulated in terms of the angle between two vectors. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>∈</m:mo> <m:mi mathvariant="script">B</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi mathvariant="script">K</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:math> Tᵢ,Sᵢ,Rᵢ∈ B(K) , where R i ’s have the polar decomposition R i = V i | R i | for i = 1, 2, …, n . Suppose that ϕ and ψ are non-negative functions satisfying ϕ ( s ) ψ ( s ) = s , ( s ≥ 0). It is shown that, if either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mn>0</m:mn> <m:mo>≤</m:mo> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>≤</m:mo> <m:mfrac> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> </m:math> 0≤ θ ᵢ< θ _Rᵢ,Tᵢx,Sᵢx≤ π /2 for all unit vectors <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mi>x</m:mi> <m:mo>∈</m:mo> <m:mi mathvariant="script">K</m:mi> </m:math> x∈ K or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <m:mfrac> <m:mrow> <m:mi>π</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> </m:mrow> </m:mfrac> <m:mo>≤</m:mo> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:msub> <m:mrow> <m:mi>R</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>T</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mi>x</m:mi> <m:mo>,</m:mo> <m:msub> <m:mrow> <m:mi>S</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow> </m:msub> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo><</m:mo> <m:msub> <m:mrow> <m:mi>θ</m:mi> </m:mrow> <m:mrow> <m:mi>i</m:mi> </m:mrow