Abstract
The Schur convexity and Schur-geometric convexity of generalized Heronian means involving two parameters are studied, the main result is then used to obtain several interesting and significantly inequalities for generalized Heronian means.
Highlights
Throughout the paper, R denotes the set of real numbers, x x1, x2, . . . , xn denotes n-tuple n-dimensional real vector, the set of vectors can be written as
We assume that a, b ∈ R2
Janous 4 presented a weighted generalization of the above Heronian-type means, as follows:
Summary
Throughout the paper, R denotes the set of real numbers, x x1, x2, . . . , xn denotes n-tuple n-dimensional real vector , the set of vectors can be written as. Throughout the paper, R denotes the set of real numbers, x x1, x2, . Xn denotes n-tuple n-dimensional real vector , the set of vectors can be written as. Janous 4 presented a weighted generalization of the above Heronian-type means, as follows:. The following exponential generalization of Heronian means was considered by Jia and Cao in 5 , Hp Hp a, b. Several variants as well as interesting applications of Heronian means can be found in the recent papers 6–11. The weighted and exponential generalizations of Heronian means motivate us to consider a unified generalization of Heronian means 1.4 and 1.5 , as follows: Hp,w a, b. The Schur convexity, Schur-geometric convexity, and monotonicity of the generalized Heronian means Hp,w a, b are discussed. Some interesting inequalities for generalized Heronian means are obtained
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