Optimal Bounds for Neuman-Sándor Mean in Terms of the Convex Combinations of Harmonic, Geometric, Quadratic, and Contraharmonic Means

and Applied Analysis 3 Theorem 1.1. The double inequality α1H a, b 1 − α1 Q a, b < M a, b < β1H a, b ( 1 − β1 ) Q a, b 1.7 holds for all a, b > 0with a/ b if and only if α1 ≥ 2/9 0.2222 . . . and β1 ≤ 1−1/ √ 2 log 1 √ 2 0.1977 . . . . Theorem 1.2. The double inequality α2G a, b 1 − α2 Q a, b < M a, b < β2G a, b ( 1 − β2 ) Q a, b 1.8 holds for all a, b > 0with a/ b if and only if α2 ≥ 1/3 0.3333 . . . and β2 ≤ 1−1/ √ 2 log 1 √ 2 0.1977 . . . . Theorem 1.3. The double inequality α3H a, b 1 − α3 C a, b < M a, b < β3H a, b ( 1 − β3 ) C a, b 1.9 holds for all a, b > 0 with a/ b if and only if α3 ≥ 1 − 1/ 2 log 1 √ 2 0.4327 . . . and β3 ≤ 5/12 0.4166 . . . . 2. Lemmas In order to prove our main results we need two Lemmas, which we present in this section. Lemma 2.1 see 21, Lemma 1.1 . Suppose that the power series f x ∑∞ n 0 anx n and g x ∑∞ n 0 bnx n have the radius of convergence r > 0 and bn > 0 for all n ∈ {0, 1, 2, . . .}. Let h x f x / g x , then the following statements are true. 1 If the sequence {an/bn}n 0 is (strictly) increasing (decreasing), then h x is also (strictly) increasing (decreasing) on 0, r . 2 If the sequence {an/bn} is (strictly) increasing (decreasing) for 0 < n ≤ n0 and (strictly) decreasing (increasing) for n > n0, then there exists x0 ∈ 0, r such that h x is (strictly) increasing (decreasing) on 0, x0 and (strictly) decreasing (increasing) on x0, r . Lemma 2.2. Let p ∈ 0, 1 , λ0 1 − 1/ √ 2 log 1 √ 2 0.1977 . . . and fp x sinh −1 x − x √ 1 x2 − p (√ 1 x2 − √ 1 − x2 ) . 2.1 Then f1/3 x < 0 and fλ0 x > 0 for all x ∈ 0, 1 . Proof. From 2.1 , one has fp 0 0, 2.2 fp 1 log ( 1 √ 2 ) − 1 √ 2 ( 1 − p , 2.3 f ′ p x gp x √ 1 − x4 (√ 1 x2 p (√ 1 − x2 − √ 1 x2 ))2 , 2.4 4 Abstract and Applied Analysis