A New Method to Study Analytic Inequalities
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Research Article A New Method to Study Analytic Inequalities Xiao-Ming Zhang and Yu-Ming Chu Department of Mathematics, Huzhou Teachers College, Huzhou 313000, China Correspondence should be addressed to Yu-Ming Chu, [email protected] Received 16 October 2009; Accepted 24 December 2009 Academic Editor: Kunquan Lan Copyright q 2010 X.-M. Zhang and Y.-M. Chu. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. We present a new method to study analytic inequalities involving n variables. Regarding its applications, we proved some well-known inequalities and improved Carleman’s inequality.
1. Monotonicity Theorems Throughout this paper, we denote R the set of real numbers and R the set of strictly positive real numbers, n ∈ N, n ≥ 2. In this section, we present the main results of this paper. Theorem 1.1. Suppose that a, b ∈ R with a < b and c ∈ a, b, f : a, bn → R has continuous partial derivatives and c , Dm x1 , x2 , . . . , xn−1 , c | min {xk } ≥ c, xm max {xk } / 1≤k≤n−1
1≤k≤n−1
m 1, 2, . . . , n − 1. 1.1
If ∂fx/∂xm > 0 for all x ∈ Dm m 1, 2, . . . , n − 1, then f y1 , y2 , . . . , yn−1 , c ≥ fc, c, . . . c, c,
for all ym ∈ c, b m 1, 2, . . . , n − 1.
1.2
2
Journal of Inequalities and Applications
Proof. Without loss of generality, since we assume that n 3 and y1 > y2 > c. For x1 ∈ y2 , y1 , we clearly see that x1 , y2 , c ∈ D1 , then ∂fx > 0. ∂x1 xx1 ,y2 ,c
1.3
From the continuity of the partial derivatives of f and ∂fx > 0, ∂x1 xy2 ,y2 ,c
1.4
we know that there exists ε > 0 such that y2 − ε ≥ c and ∂fx > 0, ∂x1 xx1 ,y2 ,c
1.5
for any x1 ∈ y2 − ε, y2 . Hence, since f·, y2 , c : x1 ∈ y2 − ε, y1 → fx1 , y2 , c is strictly monotone increasing, then we have f y1 , y2 , c > f y2 , y2 , c > f y2 − ε, y2 , c .
1.6
Next, for x2 ∈ y2 − ε, y2 , then y2 − ε, x2 , c ∈ D2 and ∂fx > 0. ∂x2 xy2 −ε,x2 ,c
1.7
f y1 , y2 , c > f y2 , y2 , c > f y2 − ε, y2 , c > f y2 − ε, y2 − ε, c .
1.8
Hence, we get
If y2 − ε c, then Theorem 1.1 is true. Otherwise, we repeat the above process and we clearly see that the first and second variables in f are decreasing and no less than c. Let s, t be their limit values, respectively, then fy1 , y2 , c > fs, t, c and s, t ≥ c. If s c, t c, then Theorem 1.1 is also true; otherwise, we repeat the above process again and denote p and q the greatest lower bounds for the first and the second variables , respectively. We clearly see that p q c; therefore, fy1 , y2 , c > fc, c, c and Theorem 1.1 is true. Similarly, we have the following theorem.
Journal of Inequalities and Applications
3
Theorem 1.2. Suppose that a, b ∈ R with a < b and c ∈ a, b, f : a, bn → R has continuous partial derivatives and Em
c , x1 , x2 , . . . , xn−1 , c | max {xk
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