A generalization of Mulholland's inequality (Q1415206)

Mulholland's inequality states that if \(p > 1, \frac1{p} +\frac1{q} = 1, \left\{a_ n\right\}\) and \(\left\{b_ n\right\}\) are nonnegative sequences of real numbers such that \(0 < \sum^{\infty}_{n=2}\frac1{n}a^ p_ n < \infty\) and \(0 < \sum^{\infty}_{n=2}\frac1{n}b^ q_ n < \infty,\) then \[ \sum^{\infty}_{n=2}\sum^{\infty}_{m=2}\frac{a_ m b_ n}{m+n} < \frac{\pi}{\sin\Big(\frac{\pi}{p}\Big)} \left\{\sum^{\infty}_{n=1}a^ p_ n\right\}^{\frac1{p}} \left\{\sum^{\infty}_{n=1}b^ q_ n\right\}^{\frac1{q}}. \] In this paper, the authors generalize the above result which was obtained by \textit{H. P. Mulholland} [J. Lond. Math. Soc. 6, 100--106 (1931; Zbl 0001.33203)] by introducing three parameters \(r,s\) and \(\lambda\) with the best constant factor involving the \(\beta\) function.