A New Proof Method of Analytic Inequality

This paper gives a new proof method of analytic inequality involving n variables. As its Applications, we proved some well-known inequalities and improved the Carleman-Inequality. 1. monotonicity on special region Throughout the paper R denotes the set of real numbers and R+ denotes the set of strictly positive real numbers, n ∈ N, n ≥ 2. In this section, we shall provide a new proof method of analytic inequality involving n variables. Theorem 1.1. Given a, b ∈ R, c ∈ [a, b]. Let f : x ∈ [a, b] → R have continuous partial derivative, Di = { (x1, x2, · · · , xn−1, c) | min 1≤k≤n−1 {xk} ≥ c, xi = max 1≤k≤n−1 {xk} 6 = c } , i = 1, 2, · · · , n− 1. If ∂f (x)/∂xi > 0 hold for any x ∈ Di (i = 1, 2, · · · , n− 1), then f (y1, y2, · · · , yn−1, c) ≥ f (c, c, · · · c, c) hold for yi ∈ [c, b] (i = 1, 2, · · · , n− 1). Proof. Without the losing of generality, we let n = 3 and y1 > y2 > c. For x1 ∈ [y2, y1], it has (x1, y2, c) ∈ D1, then ∂f (x)/∂x1|x=(x1,y2,c) > 0. Owing to the continuity of partial derivative and ∂f (x)/∂x1|x=(y2,y2,c) > 0, it exists ε, such that y2 − ε ≥ c and ∂f (x)/∂x1|x=(x1,y2,c) > 0 for any x1 ∈ [y2 − ε, y2]. Hence, f (·, y2, c) : x1 ∈ [y2 − ε, y1] → f (x1, y2, c) is strictly monotone increasing, f (y1, y2, c) > f (y2, y2, c) > f (y2 − ε, y2, c) . For x2 ∈ [y2 − ε, y2], (y2 − ε, x2, c) ∈ D2, ∂f (x)/∂x2|x=(y2−ε,x2,c) > 0. Then f (y1, y2, c) > f (y2, y2, c) > f (y2 − ε, y2, c) > f (y2 − ε, y2 − ε, c) . If y2− ε = c, this completes the proof of the Theorem 1.1. Otherwise, we repeat the above process. It is clear that the first variable and the second variable of function f are decreasing and no less than c. Let s, t are their limits, then f (y1, y2, c) > f (s, t, c), where s, t ≥ c. If s = c, t = c, this completes the proof of the Theorem 1.1. Otherwise, we repeat the above process. Let the greatest lower bound of the first variable and the second variable are p, q. It is easy to see p = q = c, and f (y1, y2, c) > f (c, c, c). Similarly to the above ,we know Theorem 1.2 is true. Theorem 1.2. Given a, b ∈ R, c ∈ [a, b]. Let f : x ∈ [a, b] → R have continuous partial derivative, Di = { (x1, x2, · · · , xn−1, c) | max 1≤k≤n−1 {xk} ≤ c, xi = min 1≤k≤n−1 {xk} 6 = c } , i = 1, 2, · · · , n− 1. Date: January 28, 2009. 2000 Mathematics Subject Classification. Primary 26A48, 26B35, 26D20,