Some Refinements and Generalizations of I. Schur Type Inequalities

Abstract

Recently, extensive researches on estimating the value of e have been studied. In this paper, the structural characteristics of I. Schur type inequalities are exploited to generalize the corresponding inequalities by variable parameter techniques. Some novel upper and lower bounds for the I. Schur inequality have also been obtained and the upper bounds may be obtained with the help of Maple and automated proving package (Bottema). Numerical examples are employed to demonstrate the reliability of the approximation of these new upper and lower bounds, which improve some known results in the recent literature.

The construction of several new inequalities for definite integral

“…where M n,1 is defined in Theorem 7. Likewise, the equality holds in(8) if and only if f(x) = µ(x − c) n+1 , i.e., f (x) = µ n+2 (x − c) n+2 + C. Noting that f (c) = C and f (a) = f (b) one may get f (x) = f (c)+( f (a) − f (c)) x−c a−cn+2 , and this completes the proof. The condition f (a) = f (b) and n is an odd number in Theorem 7 can be replaced by f (a) = f (b) = 0 and n ∈ , while the result remains the same.…”

“…In 2005, Yang et al [7] established an extension on Hardy-Hilbert integral inequality by introducing a power exponent function, and show the best possible coefficient. In 2014, Gu et al [8] improved the upper and lower bounds for the I. Schur inequality, and obtained some new I. Schur inequality. As to the mathematical equivalence among some famous inequalities, Li et al [9] presented several generalization of the Radon inequality, and proved the equivalence relation of the weighted power mean inequality and Radon inequality.…”

The Weighted Arithmetic Mean–Geometric Mean Inequality is Equivalent to the Hölder Inequality

“…[2, pp. 74-75, Theorem 7.6]) is often inferred from the Jensen inequality, which is a more generalized inequality than AM-GM inequality, refer to [1][2][3] and references therein. In addition, the well-known Hölder inequality [3], found by Rogers (1888) and discovered independently by Otto Hölder (1889), is a basic inequality between integrals and an indispensable tool for the study of L p space, and is a extension form of Cauchy-Bunyakovsky-Schwarz inequality [4], Hölder inequality is used to prove the Minkowski inequality, which is the triangle inequality (refer to [5][6][7][8]). Weighted power means (also known as generalized means) M m r (a) for a sequence a = (a 1 , a 2 , .…”