A New Hilbert-Type Inequality with Positive Homogeneous Kernel and Its Equivalent Forms

• Main Authors: Bicheng Yang, Shanhe Wu, Aizhen Wang
• Format: Article
• Language: English
• Published: MDPI AG 2020-02-01
• Series: Symmetry
• Subjects: hilbert-type inequality; homogeneous kernel; equivalent statement; operator expression; euler–maclaurin summation formula
• Online Access: https://www.mdpi.com/2073-8994/12/3/342
• ISSN: 2073-8994

We establish a new inequality of Hilbert-type containing positive homogeneous kernel <inline-formula> <math display="inline"> <semantics> <mrow> <msup> <mrow> <mo stretchy="false">(</mo> <mi>min</mi> <mo>{</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>}</mo> <mo stretchy="false">)</mo> </mrow> <mi>&#955;</mi> </msup> </mrow> </semantics> </math> </inline-formula> and derive its equivalent forms. Based on the obtained Hilbert-type inequality, we discuss its equivalent forms and give the operator expressions in some particular cases.