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In this paper, we prove a further generalized refinement of the weighted arithmetic-geometric mean inequality. As application, we show a new refinement of the generalized classical Hölder’s inequality and we give refinements to several inequalities for some special functions.

In this paper, we prove that if a, b > 0 and 0 ≤ v ≤ 1. Then for all positive integer m (1) - For v ∈ v ∈ [ 0 , 1 2 n ] v \in \left[ {0,{1 \over {{2^n}}}} \right] , we have ( a v b 1 - v ) m + ∑ k = 1 n 2 k - 1 v m ( b m - ( a b 2 k - 1 - 1 ) m 2 k ) 2 ≤ ( v a + ( 1 - v ) b ) m . {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{v^m}{{\left( {\sqrt {{b^m}} - \root {{2^k}} \of {\left( {a{b^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m}.} (2) - For v ∈ v ∈ [ 2 n - 1 2 n , 1 ] v \in \left[ {{{{2^n} - 1} \over {{2^n}}},1} \right] , we have ( a v b 1 - v ) m + ∑ k = 1 n 2 k - 1 ( 1 - v ) m ( a m - ( b a 2 k - 1 - 1 ) m 2 k ) 2 ≤ ( v a + ( 1 - v ) b ) m , {\left( {{a^v}{b^{1 - v}}} \right)^m} + \sum\limits_{k = 1}^n {{2^{k - 1}}{{\left( {1 - v} \right)}^m}{{\left( {\sqrt {{a^m}} - \root {{2^k}} \of {\left( {b{a^{2k - 1}} - 1} \right)m} } \right)}^2} \le {{\left( {va + \left( {1 - v} \right)b} \right)}^m},} we also prove two similar inequalities for the cases v ∈ v ∈ [ 2 n - 1 2 n , 1 2 ] v \in \left[ {{{{2^n} - 1} \over {{2^n}}},{1 \over 2}} \right] and v ∈ v ∈ [ 1 2 , 2 n + 1 2 n ] v \in \left[ {{1 \over 2},{{{2^n} + 1} \over {{2^n}}}} \right] . These inequalities provides a generalization of an important refinements of the Young inequality obtained in 2017 by S. Furuichi. As applications we shall give some refined Young type inequalities for the traces, determinants, and p-norms of positive τ-measurable operators.

We establish some new refinements to the Hölder inequality. We then apply them to provide some refinements to the extended Euler’s gamma and beta functions. As another application of our results, we give a new proof of the equivalence between the Hölder inequality and the Cauchy-Schwarz inequality.

The aim of this paper is to give some refinements to several inequalities, recently etablished, by P.K. Bhandari and S.K. Bissu in [Inequalities via Hölder’s inequality, Scholars Journal of Research in Mathematics and Computer Science, 2 (2018), no. 2, 124–129] for the incomplete gamma function, Polygamma functions, Exponential integral function, Abramowitz function, Hurwitz-Lerch zeta function and for the normalizing constant of the generalized inverse Gaussian distribution and the Remainder of the Binet’s first formula for ln Γ(x).

In this work, we give further refinements of one of the most important extensions to Young’s inequalities due to Alzer–Fonseca–Kovacec (Linear Multilinear Algebra 63(3):622–635, 2015). As applications, we show some related inequalities for operators and some refinements of Young-type inequalities for determinants of positive $$\tau $$ -measurable operators.

In this work, we give a multiple-term refinement of Young’s inequality which allow us to generalize and unify several results. As applications, we provide further refinements of a reversed AM–GM operator inequalities which extends and unifies two recent and important results due to Yang et al. (Math Slovaca 69:919–930, 2019) and Ren et al. (J Inequal Appl 2020:98, 2020) for positive linear maps and matrices. Also, our work deals with several other related results to both scalar and operator versions of the generalized Young’s inequality. In particular, we give a multiple-term refinement of Young’s inequalities for Hilbert–Schmidt norm, the determinants and the traces of positive definite matrices.

In this paper, we show a new generalized refinement of Young's inequality. As applications we give some new generalized refinements of Young type inequalities for the traces, determinants, and norms of positive definite matrices.