Some results of Young-type inequalities

In this paper, one of our main targets is to present some improvements of Young-type inequalities due to Alzer et al. (Linear Multilinear Algebra 63(3):622-635, 2015) under some conditions. That is to say: when 0 < nu, tau < 1, a, b > 0, we have a del(nu)b - a#(nu)b/a del(tau)b - a#(tau)b <= nu(1 - nu)/tau(1 - tau) and (a del(nu)b)(2) - (a#(nu)b)(2)/(a del(tau)b)(2) - (a#(tau)b)(2) <= nu(1 - nu)/tau(1 - tau) for (b - a)( tau -nu) >= 0; and the inequalities are reversed if (b - a)(tau - nu) <= 0. In addition, we show a new Young-type inequality (1 - v(N+1) + v(N+2))a + (1 - v(2))b <= v(vN-(N+1))a(v)b(1-v) + (root a - root b)(2) for 0 <= nu <= 1, N is an element of N and a, b > 0. Then we can get some related results about operators, Hilbert-Schmidt norms, determinants by these scalars results.