Conjugate Exponents Research Articles

The Hilbert-Type Integral Inequalities with a Homogeneous Kernel of -Degree

By introducing an integral operator, a norm with a weight function, and two pairs of conjugate exponents, we find the conditions for the Hilbert-type integral inequalities with a homogeneous kernel of -degree. We also prove that the constant factors in the inequalities are all the best possible. As the particular situations, some new inequalities with a homogeneous kernel and their other two forms are given. We extend some previous results.

On Hilbert's type inequalities

By introducing two pairs of conjugate exponents and estimating the weight coefficients, we give generalizations of some Hilbert's type inequalities with the best constant factor. As applications, some particular results are considered.

On a Multiple Hilbert-Type Integral Inequality with the Symmetric Kernel

We build a multiple Hilbert-type integral inequality with the symmetric kernel and involving an integral operator . For this objective, we introduce a norm , two pairs of conjugate exponents and , and two parameters. As applications, the equivalent form, the reverse forms, and some particular inequalities are given. We also prove that the constant factors in the new inequalities are all the best possible.

Bohr's inequalities for Hilbert space operators

The classical Bohr's inequality states that | z + w | 2 ⩽ p | z | 2 + q | w | 2 for all z , w ∈ C and all p , q > 1 with 1 p + 1 q = 1 . In this paper, Bohr's inequality is generalized to the context of Hilbert space operators for all positive conjugate exponents p , q ∈ R . In particular, the parallelogram law is recovered and some other interesting operator inequalities and established.