* Correspondence: qlhuang@yeah. net Department of Mathematics, Guangdong University of Education, Guangzhou, Guangdong 510303, People’s Republic of China Abstract By using the way of weight coefficients and the idea of introducing parameters and by means of Hadamard’s inequality, we give a more accurate half-discrete Hilbert’s inequality with a best constant factor. We also consider its best ex...

By introducing several independent parameters, according to the structural symmetry of quasi-homogeneous kernels and Hilbert-type inequality, using weight function method, parameter conditions optimal n-multiple series inequality with are discussed, equivalent expression formula best constant factor obtained. As applications, some special symmetric inequalities given.

\begin{abstract} In this paper we address the problem of finding best constants in inequalities form: $$ \|\big(|P_+f|^s+|P_-f|^s\big)^{\frac{1}{s}}\|_{L^p({\mathbb{T}})}\leq A_{p,s} \|f\|_{L^p({\mathbb{T}})},$$ where $P_+f$ and $P_-f$ denote analytic co-analytic projection a complex-valued function $f \in L^p({\mathbb{T}}),$ for $p \geq 2$ all $s>0$, thus proving Hollenbeck-Verbitsky conjectur...