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<p>This study establishes a complete characterization of convexity for the function <Equation ID="Equ5"> <EquationSource Format="TEX">\(\begin{aligned} K_{p}(x)=(x^{\top }A^{-p}x)(x^{\top }Ax)^{p} \end{aligned}\)</EquationSource> </Equation>in 2-dimensional spaces, where <i>A</i> is a real positive definite matrix, and <InlineEquation ID="IEq5"> <EquationSource Format="TEX">\(p&gt;0\)</EquationSource> </InlineEquation>. We prove that <InlineEquation ID="IEq6"> <EquationSource Format="TEX">\(K_{p}(x)\)</EquationSource> </InlineEquation> is convex if and only if <Equation ID="Equ6"> <EquationSource Format="TEX">\(\begin{aligned} \frac{\lambda _{\max }(A)}{\lambda _{\min }(A)}\le \left( \frac{4p^{2} +9p+4+2\sqrt{2}(p+1)\sqrt{2p^{2}+5p+2}}{p}\right) ^{\frac{1}{1+p}}. \end{aligned}\)</EquationSource> </Equation>This generalizes Zhao’s result (J Comput Appl Math 235(15):4389–4403, 2011), which addressed the specific case <InlineEquation ID="IEq7"> <EquationSource Format="TEX">\(p=1,\)</EquationSource> </InlineEquation> to all positive values <i>p</i> in 2-dimensional spaces.</p>

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A necessary and sufficient condition for the convexity of \(\left( x^{T} A^{-p}x\right) \left( x^{T}Ax\right) ^{p}\) for \(p>0\) in 2-dimensional spaces

  • 期刊论文

  • 发表日期：

    2025年9月24日

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  • Ibrahim Halil Gumus

摘要

This study establishes a complete characterization of convexity for the function \(\begin{aligned} K_{p}(x)=(x^{\top }A^{-p}x)(x^{\top }Ax)^{p} \end{aligned}\) in 2-dimensional spaces, where A is a real positive definite matrix, and \(p>0\) . We prove that \(K_{p}(x)\) is convex if and only if \(\begin{aligned} \frac{\lambda _{\max }(A)}{\lambda _{\min }(A)}\le \left( \frac{4p^{2} +9p+4+2\sqrt{2}(p+1)\sqrt{2p^{2}+5p+2}}{p}\right) ^{\frac{1}{1+p}}. \end{aligned}\) This generalizes Zhao’s result (J Comput Appl Math 235(15):4389–4403, 2011), which addressed the specific case \(p=1,\) to all positive values p in 2-dimensional spaces.