Certain Inequalities Related to the Generalized Numeric Range and Numeric Radius That Are Associated with Convex Functions

In this paper, we delve into the intricate connections between the numerical ranges of specific operators and their transformations using a convex function. Furthermore, we derive inequalities related to the numerical radius. These relationships and inequalities are built upon well-established principles of convexity, which are applicable to non-negative real numbers and operator inequalities. To be more precise, our investigation yields the following outcome: consider the operators <a:math xmlns:a="http://www.w3.org/1998/Math/MathML" id="M1"><a:mi>A</a:mi></a:math> and <c:math xmlns:c="http://www.w3.org/1998/Math/MathML" id="M2"><c:mi>B</c:mi></c:math> , both of which are positive and have spectra within the interval <e:math xmlns:e="http://www.w3.org/1998/Math/MathML" id="M3"><e:mfenced open="[" close="]" separators="|"><e:mrow><e:mi>m</e:mi><e:mo>,</e:mo><e:mi>M</e:mi></e:mrow></e:mfenced></e:math> , denoted as <j:math xmlns:j="http://www.w3.org/1998/Math/MathML" id="M4"><j:mi>σ</j:mi><j:mfenced open="(" close=")" separators="|"><j:mrow><j:mi>A</j:mi></j:mrow></j:mfenced></j:math> and <o:math xmlns:o="http://www.w3.org/1998/Math/MathML" id="M5"><o:mi>σ</o:mi><o:mfenced open="(" close=")" separators="|"><o:mrow><o:mi>B</o:mi></o:mrow></o:mfenced></o:math> . In addition, let us introduce two monotone continuous functions, namely, <t:math xmlns:t="http://www.w3.org/1998/Math/MathML" id="M6"><t:mi>g</t:mi></t:math> and <v:math xmlns:v="http://www.w3.org/1998/Math/MathML" id="M7"><v:mi>h</v:mi></v:math> , defined on the interval <x:math xmlns:x="http://www.w3.org/1998/Math/MathML" id="M8"><x:mfenced open="[" close="]" separators="|"><x:mrow><x:mi>m</x:mi><x:mo>,</x:mo><x:mi>M</x:mi></x:mrow></x:mfenced></x:math> . Let <cb:math xmlns:cb="http://www.w3.org/1998/Math/MathML" id="M9"><cb:mi>f</cb:mi></cb:math> be a positive, increasing, convex function possessing a supermultiplicative property, which means that for all real numbers <eb:math xmlns:eb="http://www.w3.org/1998/Math/MathML" id="M10"><eb:mi>t</eb:mi></eb:math> and <gb:math xmlns:gb="http://www.w3.org/1998/Math/MathML" id="M11"><gb:mi>s</gb:mi></gb:math> , we have <ib:math xmlns:ib="http://www.w3.org/1998/Math/MathML" id="M12"><ib:mi>f</ib:mi><ib:mfenced open="(" close=")" separators="|"><ib:mrow><ib:mtext>ts</ib:mtext></ib:mrow></ib:mfenced><ib:mo>≤</ib:mo><ib:mi>f</ib:mi><ib:mfenced open="(" close=")" separators="|"><ib:mrow><ib:mi>t</ib:mi></ib:mrow></ib:mfenced><ib:mi>f</ib:mi><ib:mfenced open="(" close=")" separators="|"><ib:mrow><ib:mi>s</ib:mi></ib:mrow></ib:mfenced></ib:math> . Under these specified conditions, we establish the following inequality: for all <tb:math xmlns:tb="http://www.w3.org/1998/Math/MathML" id="M13"><tb:mn>0</tb:mn><tb:mo>≤</tb:mo><tb:mi>ν</tb:mi><tb:mo>≤</tb:mo><tb:mn>1</tb:mn></tb:math> , this outcome highlights the intricate relationship between the numerical range of the expression <vb:math xmlns:vb="http://www.w3.org/1998/Math/MathML" id="M14"><vb:msup><vb:mrow><vb:mi>g</vb:mi></vb:mrow><vb:mrow><vb:mi>ν</vb:mi></vb:mrow></vb:msup><vb:mfenced open="(" close=")" separators="|"><vb:mrow><vb:mi>A</vb:mi></vb:mrow></vb:mfenced><vb:mi>X</vb:mi><vb:msup><vb:mrow><vb:mi>h</vb:mi></vb:mrow><vb:mrow><vb:mn>1</vb:mn><vb:mo>−</vb:mo><vb:mi>ν</vb:mi></vb:mrow></vb:msup></vb:math> when transformed by the convex function <ac:math xmlns:ac="http://www.w3.org/1998/Math/MathML" id="M15"><ac:mi>f</ac:mi></ac:math> and the norm of <cc:math xmlns:cc="http://www.w3.org/1998/Math/MathML" id="M16"><cc:mi>X</cc:mi></cc:math> . Importantly, this inequality holds true for a broad range of values of <ec:math xmlns:ec="http://www.w3.org/1998/Math/MathML" id="M17"><ec:mi>ν</ec:mi></ec:math> . Furthermore, we provide supportive examples to validate these results.