| Summary: | Finding the range of coordinated convex functions is yet another application for the symmetric Hermite–Hadamard inequality. For any two-dimensional interval <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mrow><mo>[</mo><msub><mi mathvariant="normal">a</mi><mn>0</mn></msub><mo>,</mo><mspace width="0.166667em"></mspace><msub><mi mathvariant="normal">a</mi><mn>1</mn></msub><mo>]</mo></mrow><mo>×</mo><mrow><mo>[</mo><msub><mi mathvariant="normal">c</mi><mn>0</mn></msub><mo>,</mo><mspace width="0.166667em"></mspace><msub><mi mathvariant="normal">c</mi><mn>1</mn></msub><mo>]</mo></mrow><mo>⊆</mo><msup><mo>ℜ</mo><mn>2</mn></msup></mrow></semantics></math></inline-formula>, we introduce the notion of partial <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">q</mi><mi>θ</mi></msub></semantics></math></inline-formula>-, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msub><mi mathvariant="normal">q</mi><mi>ϕ</mi></msub></semantics></math></inline-formula>-, and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">q</mi><mi>θ</mi></msub><msub><mi mathvariant="normal">q</mi><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula>-symmetric derivatives and a <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">q</mi><mi>θ</mi></msub><msub><mi mathvariant="normal">q</mi><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula>-symmetric integral. Moreover, we will construct the <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi mathvariant="normal">q</mi><mi>θ</mi></msub><msub><mi mathvariant="normal">q</mi><mi>ϕ</mi></msub></mrow></semantics></math></inline-formula>-symmetric Hölder’s inequality, the symmetric quantum Hermite–Hadamard inequality for the function of two variables in a rectangular plane, and address some of its related applications.
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