On Certain Inequalities for Neuman-Sándor Mean

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) = ∑ ∞ n=0 a n x n and g(x) = ∑∞ n=0 b n x n have the radius of convergence r > 0 and b n > 0 for all n ∈ {0, 1, 2, . . .}. Let h(x) = f(x)/g(x). Then, (1) if the sequence {a n /b n } ∞ n=0 is (strictly) increasing (decreasing), then h(x) is also (strictly) increasing (decreasing) on (0, r); (2) if the sequence {a n /b n } is (strictly) increasing (decreasing) for 0 < n ≤ n 0 and (strictly) decreasing (increasing) for n > n 0 , then there existsx 0 ∈ (0, r) such that h(x) is (strictly) increasing (decreasing) on (0, x 0 ) and (strictly) decreasing (increasing) on (x 0 , r). Lemma 7. The function h (t) = t cosh (3t) + 11t cosh (t) − sinh (3t) − 9 sinh (t) 2t [cosh (3t) − cosh (t)] (12) is strictly decreasing on (0, log(1 + √2)), where sinh(t) = (et − e −t )/2 and cosh(t) = (et + e−t)/2 are the hyperbolic sine and cosine functions, respectively. Proof. Let h 1 (t) = t cosh (3t) + 11t cosh (t) − sinh (3t) − 9 sinh (t) , h 2 (t) = 2t [cosh (3t) − cosh (t)] . (13) Then, making use of power series formulas, we have h 1 (t) = t ∞ ∑ n=0 (3t) 2n (2n)! + 11t ∞ ∑ n=0 (t) 2n