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The Scientific World Journal
Volume 2013 (2013), Article ID 645201, 4 pages
On Satnoianu-Wu’s Inequality
College of Mathematics, Inner Mongolia University for Nationalities, Tongliao City, Inner Mongolia 028043, China
Received 22 April 2013; Accepted 28 June 2013
Academic Editors: F. J. Garcia-Pacheco, Y. Sawano, and S. Tikhonov
Copyright © 2013 Bo-Yan Xi. 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.
By applying techniques in the theory of convex functions and Schur-geometrically convex functions, the author investigates a conjecture of Satnoianu on an algebraic inequality and generalizes some known results in recent years.
For and with , let These quantities are, respectively, called the arithmetic, geometric, and harmonic means of a positive sequence . For more information on the theory of means, see the monograph  or the papers [2–5] and plenty of references therein.
For convenience, in what follows, we use the notation for and .
In , Satnoianu posed the following conjecture.
Conjecture 1. For , , and , it is valid that
Theorem 2 (see [10, Theorem 1]). Let be a positive integer and let , , and for be positive numbers. If and then
The goal of this paper is to answer these questions.
2. Definitions and Lemmas
We need the following definitions and lemmas.
Definition 3 (see [11, page 8]). Let and . We say that is majorized by (in symbols ) if for , where and are rearrangements of and in a descending order.
Definition 4 (see ). Let .(1)The set is said to be geometrically convex if for every and .(2)A function is said to be Schur-geometrically convex on if implies for every .(3)A function is said to be Schur-geometrically concave on if implies for every .
Lemma 5 (see ). Let be a symmetric and geometrically convex set with inner points and a symmetric and differentiable function in . Then is a Schur-geometrically convex (or Schur-geometrically concave, resp.) function on if and only if
3. Main Results
Now we start off to demonstrate our main results.
Proof. For and with , since
then(1)for , or for and , the function is convex on ;(2)for , or for and , the function is concave on .By Jensen’s inequality, when , or when and , we have
where . When , or when and , inequality (11) is reversed.
Let for , , , and . Then it is easy to show that . Making use of (11), we obtain inequality (9). The proof of Theorem 7 is complete.
Corollary 9. Let , for , and . If , then if , inequality (13) is reversed.
Proof. This follows from Theorem 7 by considering that if , we have and that if , we have .
Proof. For , let
for . When , or when and , we have
when and , inequality (17) is reversed.
Using Lemma 5, we have the following conclusions:(1)if , or if and , the function is Schur-geometrically convex on ;(2)if and , the function is Schur-geometrically concave on .By the fact that and by Definition 4, if , or if and , we have if and , inequality (19) is reversed.
Letting for leads to . Putting , and in inequality (19) results in inequality (14). The proof of Theorem 10 is complete.
Theorem 13. Let , for , and with . If , then if , inequality (22) is reversed.
Proof. Since is a convex (or concave, resp.) function on for or (or for , resp.), by Jensen's inequality, if or , we have
if , inequality (23) is reversed.
Letting and shows . Further from (23), we obtain inequality (22). The proof of Theorem 13 is complete.
Corollary 15. Let , for , with , and defined as in (2).(1)When , one has (2)When and , one has (3)When and , one has (4)When and , one has
Corollary 16. Under the conditions of Corollary 15 and when ,(1)if and , one has (2)if and , one has (3)if and , one has
The author thanks four anonymous referees for their careful corrections to and valuable comments on the original version of this paper. This work was partially supported by the Foundation of the Research Program of Science and Technology at Universities of Inner Mongolia Autonomous Region under Grant no. NJZY13159, China.
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