Wei-Mao Qian

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and Applied Analysis 3 (a, b), and let g󸀠(x) ̸ = 0 on (a, b). If f󸀠(x)/g󸀠(x) is increasing (decreasing) on (a, b), then so are f (x) − f (a) g (x) − g (a) , f (x) − f (b) g (x) − g (b) . (11) If f󸀠(x)/g󸀠(x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 6 (see [11, Lemma 1.1]). Suppose that the power series f(x) = ∑(More)
and Applied Analysis 3 If f(x)/g(x) is strictly monotone, then the monotonicity in the conclusion is also strict. Lemma 2. Let u, α ∈ (0, 1) and f u,α (x) = ux 2 − (1 − α) ( x arctanx − 1) . (12) Then f u,α (x) > 0 for all x ∈ (0, 1) if and only if u ≥ (1 − α)/3 andf u,α (x) < 0 for allx ∈ (0, 1) if and only if u ≤ (1−α)(4/π− 1). Proof. From (12), one has f(More)
Ying-Qing Song, Wei-Mao Qian, Yun-Liang Jiang, and Yu-Ming Chu 1 School of Mathematics and Computation Sciences, Hunan City University, Yiyang, Hunan 413000, China 2 School of Distance Education, Huzhou Broadcast and TV University, Huzhou, Zhejiang 313000, China 3 School of Information & Engineering, Huzhou Teachers College, Huzhou, Zhejiang 313000, China(More)
*Correspondence: chuyuming2005@126.com 2School of Mathematics and Computation Science, Hunan City University, Yiyang, 413000, China Full list of author information is available at the end of the article Abstract In this paper, we present sharp bounds for the two Neuman means SHA and SCA derived from the Schwab-Borchardt mean in terms of convex combinations(More)
In the paper, we find the greatest values α1, α2, α3, α4 and the least values β1, β2, β3, β4 such that the double inequalities α1A(a, b) + (1− α1)H(a, b) < N ( A(a, b), G(a, b) ) < β1A(a, b) + (1− β1)H(a, b), α2A(a, b) + (1− α2)H(a, b) < N ( G(a, b), A(a, b) ) < β2A(a, b) + (1− β2)H(a, b), α3C(a, b) + (1− α3)A(a, b) < N ( Q(a, b), A(a, b) ) < β3C(a, b) +(More)
For p ∈ R, the generalized logarithmic mean Lp a, b , arithmetic mean A a, b and geometric mean G a, b of two positive numbers a and b are defined by Lp a, b a, a b; Lp a, b a 1 − b 1 / p 1 a − b , p / 0, p / − 1, a/ b; Lp a, b 1/e b/a 1/ b−a , p 0, a/ b; Lp a, b b − a / ln b − lna , p −1, a/ b; A a, b a b /2 and G a, b √ ab, respectively. In this paper, we(More)
We prove that αH a, b 1 − α L a, b > M 1−4α /3 a, b for α ∈ 0, 1 and all a, b > 0 with a/ b if and only if α ∈ 1/4, 1 and αH a, b 1 − α L a, b < M 1−4α /3 a, b if and only if α ∈ 0, 3√345/80 − 11/16 , and the parameter 1 − 4α /3 is the best possible in either case. Here, H a, b 2ab/ a b , L a, b a − b / loga − log b , and Mp a, b a b /2 1/p p / 0 and M0 a,(More)