Improvements of Young inequality using the Kantorovich constant

Some improvements of Young inequality and its reverse for positiveý ýnumbers with Kantorovich constant K(tý,ý2)=(1)24tý ýare givený. ýUsing these inequalities some operator inequalities andý ýHilbert-Schmidt norm versions for matrices are provedý. ýIný ýparticularý, ýit is shown that if aý,ýb are positive numbers andý ý0⩽ν⩽1, then for all integers k⩾ýý1ý:ýý ýK(h12n,2)rna♯νb⩽a∇νbý−ý∑n−1k=0rk((a♯mk2kbýý)12ý−ý(a♯mk흩ýý)12)2⩽K(h12n,2 Rna♯νb, ýwhere mk=[2kν] is the largest integer not greater thaný ý2kνý, ýr0=min{νý,ý1−ν}ý,ý k=min{2rk−1ý,ý1−2rk−1} and Rk=1−rk.