We consider T f = ∫ x1 0 ∫ x2 0 f (t1, t2)dt1dt2 and a corresponding geometric mean operator G f = exp(1/x1x2) ∫ x1 0 ∫ x2 0 log f (t1, t2)dt1dt2. E. T. Sawyer showed that the Hardy-type inequality ‖T f ‖Lq u ≤ C‖ f ‖Lp v could be characterized by three independent conditions on the weights. We give a simple proof of the fact that if the weight v is of product type, then in fact only one condit...

Let Ai and Bi be positive definite matrices for all i=1,…,m. It is shown that|||∑i=1m(Ai2♯Bi2)r|||≤|||((∑i=1mAi)rp2(∑i=1mBi)rp(∑i=1mAi)rp2)1p|||, unitarily invariant norms, where p>0 r≥1 such that rp≥1. This gives an affirmative answer to a conjecture posed by Dinh, Ahsani Tam. The preceding inequality directly leads recent result of Audenaert in 2015.

In this paper we introduce the Drazin geometric quasi-mean AdO?B = ||BAd|?A|2 for bounded conditional operator A and B in L2(?), where has closed range ? 0. addition, discuss some measure theoretic characterizations operators classes. Moreover, practical examples are provided to illustrate obtained results.