A New Method to Prove and Find Analytic Inequalities

and Applied Analysis 3 holds in Di ∩ Dj , then f a1, a2, . . . , an ≥ ≤ f A a , A a , . . . , A a 1.7 for all a a1, a2, . . . , an ∈ D, with equality if only if a1 a2 · · · an. Proof. If n 2, then Theorem 1.2 follows from Lemma 1.1 and l |a1 − a2|/2. We assume that n ≥ 3 in the next discussion. Without loss of generality, we only prove the case of ∂f/∂xi > ∂f/∂xj with i / j. If a1 a2 · · · an, then inequality 1.7 is clearly true. If max1≤j≤n{aj}/ min1≤j≤n{aj}, then without loss of generality we assume that a1 ≥ a2 ≥ · · · ≥ an−1 ≥ an. 1 If a1 > max2≤j≤n{aj} and an < min1≤j≤n−1{aj}, then a1, a2, . . . , an ∈ D1 ∩ Dn. From Lemma 1.1 and the conditions in Theorem 1.2 we know that there exist a 1 1 and a 1 n such that l a1 − a 1 1 a 1 n − an > 0, a 1 1 a2 or a 1 n an−1, and f a1, a2, a3, . . . , an ≥ f ( a 1 1 , a2, a3, . . . , a 1 n ) . 1.8 For the sake of convenience, we denote a 1 i ai, 2 ≤ i ≤ n − 1. Consequently, f a1, a2, a3, . . . , an ≥ f ( a 1 1 , a 1 2 , a 1 3 , . . . , a 1 n ) . 1.9 If a 1 1 a 1 2 · · · a 1 n , then Theorem 1.2 holds. Otherwise, for the case of a 1 1 a 1 2 > a 1 n , a 1 1 , a 1 2 , a 1 3 , . . . , a 1 n ∈ D1 ∩ Dn and ∂f x ∂x1 ∣ ∣ ∣ ∣ x a 1 1 ,a 1 2 ,a 1 3 ,...,a 1 n > ∂f x ∂xn ∣ ∣ ∣ ∣ x a 1 1 ,a 1 2 ,a 1 3 ,...,a 1 n . 1.10 From the continuity of partial derivatives we know that there exists ε > 0 such that ∂f x ∂x1 ∣ ∣ ∣ ∣ x s,a 1 2 ,a 1 3 ,...,t > ∂f x ∂xn ∣ ∣ ∣ ∣ ∣ ∣ x s,a 1 2 ,a 1 3 ,...,t , 1.11 where s ∈ a 1 1 − ε, a 1 1 and t ∈ a 1 n , a 1 n ε . Denote a 2 1 a 1 1 − ε, a 2 n a 1 n ε, a 2 i a 1 i 2 ≤ i ≤ n − 1 . By Lemma 1.1, we get f ( a 1 1 , a 1 2 , a 1 3 , . . . , a 1 n ) ≥ f ( a 2 1 , a 2 2 , a 2 3 , . . . , a 2 n ) , 1.12 and a 2 2 max1≤i≤n{a 2 i }. For the case of a 1 1 > a 1 n−1 a 1 n , after a similar argument, we get inequality 1.12 with a 2 n−1 min1≤i≤n{a 2 i }. 4 Abstract and Applied Analysis Repeating the above steps, we get {a i 1 , a i 2 , . . . , a i n } i 1, 2, . . . such that ∑n j 1 a i j is a constant and {a i j } i 1, 2, . . . are monotone increasing decreasing sequences if aj ≥ ≤ A a , j 1, 2, 3, . . . , n, and f ( a 1 1 , a 1 2 , a 1 3 , . . . , a 1 n ) ≥ f ( a i 1 , a i 2 , a i 3 , . . . , a i n ) . 1.13 If there exists i ∈ N such that a i 1 a i 2 · · · a i n , then the proof of Theorem 1.2 is completed. Otherwise, we denote α infi∈N{max{a i 1 , a i 2 , . . . , a i n }}; without loss of generality, we assume that max { a ij 1 , a ij 2 , . . . , a ij n } a ij 1 −→ α, lim j→ ∞ ( a ij 1 , a ij 2 , . . . , a ij n ) α, b2, b3, . . . , bn , 1.14 where {ij} ∞ j 1 is a subsequence of N. Then from the continuity of function f , we get f a1, a2, a3, . . . , an ≥ f α, b2, b3, . . . , bn . 1.15 If α/ min{b2, b3, . . . , bn}, then we repeat the above arguments and get a contradiction to the definition of α. Hence α b2 b3 · · · bn. From α ∑n i 2 bi ∑n i 1 ai we get α b2 b3 · · · bn A a ; the proof of Theorem 1.2 is completed. 2 The proof for the case of a1 max2≤j≤n{aj} or an min1≤j≤n−1{aj} is implied in the proof of 1 . In particular, according to Theorem 1.2 the following corollary holds. Corollary 1.3. Suppose that D ⊂ R is a symmetric convex set with nonempty interior, f : D → R is a symmetric function with continuous partial derivatives, and D1 { x ∈ D | x1 max 1≤j≤n { xj } } − {x ∈ D | x1 x2 · · · xn}, D2 { x ∈ D | x2 min 1≤j≤n { xj } } − {x ∈ D | x1 x2 · · · xn}, D∗ D1 ∩ D2. 1.16 If ∂f/∂x1 > < ∂f/∂x2 holds in D∗, then f a1, a2, . . . , an ≥ ≤ f A a , A a , . . . , A a 1.17 for all a a1, a2, . . . , an ∈ D, and equality holds if and only if a1 a2 · · · an. Abstract and Applied Analysis 5 2. Comparing with Schur’s Condition The Schur convexity was introduced by I. Schur 2 in 1923; the following Definitions 2.1 and 2.2 can be found in 2, 3 . Definition 2.1. For u u1, u2, . . . un ,v v1, v2, . . . vn ∈ R, without loss of generality one assumes that u1 ≥ u2 ≥ · · · ≥ un and v1 ≥ v2 ≥ · · · ≥ vn. Then u is said to be majorized by v in symbols u ≺ v if∑ki 1 ui ≤ ∑k i 1 vi for k 1, 2, . . . , n − 1 and ∑n i 1 ui ∑n i 1 vi. Definition 2.2. Suppose that Ω ⊂ R. A real-valued function φ : Ω → R is said to be Schur convex Schur concave if u ≺ v implies that φ u ≤ ≥ φ v . Recall that the following so-called Schur’s condition is very useful for determining whether or not a given function is Schur convex or concave. Theorem 2.3 see 2, page 57 . Suppose that Ω ⊂ R is a symmetric convex set with nonempty interior intΩ. If φ : Ω → R is continuous on Ω and differentiable in intΩ, then φ is Schur convex (Schur concave) on Ω if and only if it is symmetric andand Applied Analysis 5 2. Comparing with Schur’s Condition The Schur convexity was introduced by I. Schur 2 in 1923; the following Definitions 2.1 and 2.2 can be found in 2, 3 . Definition 2.1. For u u1, u2, . . . un ,v v1, v2, . . . vn ∈ R, without loss of generality one assumes that u1 ≥ u2 ≥ · · · ≥ un and v1 ≥ v2 ≥ · · · ≥ vn. Then u is said to be majorized by v in symbols u ≺ v if∑ki 1 ui ≤ ∑k i 1 vi for k 1, 2, . . . , n − 1 and ∑n i 1 ui ∑n i 1 vi. Definition 2.2. Suppose that Ω ⊂ R. A real-valued function φ : Ω → R is said to be Schur convex Schur concave if u ≺ v implies that φ u ≤ ≥ φ v . Recall that the following so-called Schur’s condition is very useful for determining whether or not a given function is Schur convex or concave. Theorem 2.3 see 2, page 57 . Suppose that Ω ⊂ R is a symmetric convex set with nonempty interior intΩ. If φ : Ω → R is continuous on Ω and differentiable in intΩ, then φ is Schur convex (Schur concave) on Ω if and only if it is symmetric and u1 − u2 ( ∂φ ∂u1 − ∂φ ∂u2 ) ≥ ≤ 0 2.1 holds for any u u1, u2, . . . , un ∈ intΩ. It is well known that a convex function is not necessarily a Schur convex function, and a Schur convex function need not be convex in the ordinary sense either. However, under the assumption of ordinary convexity, f is Schur convex if and only if it is symmetric 4 . Although the Schur convexity is an important tool in researching analytic inequalities, but the restriction of symmetry cannot be used in dealing with nonsymmetric functions. Obviously, Theorem 1.2 is the generalization and development of Theorem 2.3; the following results in Sections 3–5 show that a large number of inequalities can be proved, improved, and found by Theorem 1.2. 3. A Proof for the Hölder Inequality Using Theorem 1.2 and Corollary 1.3, we can prove some well-known inequalities, for example, power mean inequality, Hölder inequality, and Minkowski inequality. In this section, we only prove the Hölder inequality. Proposition 3.1 Hölder inequality . Suppose that x1, x2, . . . , xn , ( y1, y2, . . . , yn ) ∈ R . 3.1 If p, q > 1 and 1/p 1/q 1, then