NNTDM 17 (2011), 4, 37–41
Schur convexity of Gnan mean for two variables V. Lokesha1 , K. M. Nagaraja2 , Naveen Kumar B.3 and Y.-D. Wu4 1
2
Department of Mathematics, Acharya institute of Technology, Bangalore-90, India e-mail:
[email protected]
Department of Mathematics, Sri Krishna Institute of Technology, Bangalore-90, India e-mail:
[email protected] 3
Department Of Mathematics, R.N.S.Institute of Technology, Bangalore-61, India e-mail:
[email protected] 4
Xinchang High School, Xinchang City, Zhejiang Province 312500, P. R. China e-mail:
[email protected]
Abstract: In this paper, the convexity and Schur convexity of the Gnan mean and its dual form in two variables are discussed. Keywords: Mean, Monotonicity, Inequality, Convexity. AMS Classification: 26D15
1
Introduction
For positive numbers a, b, let b ln b − a ln a exp −1 , b−a I = I(a, b) = a, a − b , a 6= b; L = L(a, b) = ln a − ln b a, a = b; √ a + ab + b H = H(a, b) = . 3
a < b;
(1.1)
a = b; (1.2)
(1.3)
These are respectively called the Identric, Logarithmic and Heron means. In [3, 7, 8], V. Lokesha et al. studied extensively and obtained some remarkable results on the weighted Heron mean, the weighted Heron dual mean and the weighted product type means and its monotonicities.
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In [5, 6], Zhang et al. gave the generalizations of Heron mean, similar product type means and their dual forms. For two variables, the above means as follows: 1
I(a, b; k) =
k Y (k + 1 − i)a + ib k i=1
k+1
and
1
∗
,
I (a, b; k) =
k Y (k − i)a + ib k+1 k
i=0
k
1 X k−i i H(a, b; k) = a k bk , k + 1 i=0
,
(1.4)
k
i 1 X k+1−i h(a, b; k) = a k+1 b k+1 , k i=1
(1.5)
where k is a natural number. Authors proved that H(a, b; k) and I ∗ (a, b; k) are monotone decreasing functions and h(a, b; k) and I(a, b; k) are monotone increasing functions with k and also established the following limitations. lim I(a, b; k) = lim I ∗ (a, b; k) = I(a, b),
k→+∞
k→+∞
and lim H(a, b; k) = lim h(a, b; k) = L(a, b).
k→+∞
k→+∞
In [2], V. Lokesha et al. defined the Gnan mean and its dual form for two variables. Also, they obtained some interesting properties, monotonic results and its limitations. The detailed discussion on convexity and Schur convexity can be found in [1, 4]. We recall the definitions which are essential to develop this paper.
Definition 1.1. [2] Let a > 0 and b > 0, and k be a non-negative integer, α, β two real numbers. The Gnan mean G(a, b; k, α, β) and its dual g(a, b; k, α, β) forms as below; β # β1 k 1 X (k + 1 − i)aα + ibα α G(a, b; k, α, β) = , k i=1 k+1 " k # β1 X (k+1−i)β iβ 1 G(a, b; k, 0, β) = a k+1 b k+1 , k i=1 1 k Y (k + 1 − i)aα + ibα kα , G(a, b; k, α, 0) = k+1 i=1 √ G(a, b; k, 0, 0) = ab; "
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(1.6)
and " g(a, b; k, α, β) =
β
1 k+1
"
k X (k − i)aα + ibα α k # β1
i=0 k
# β1 ,
1 X (k−i)β iβ , g(a, b; k, 0, β) = a k bk k + 1 i=0 1 k Y (k − i)aα + ibα (k+1)α g(a, b; k, α, 0) = , k i=0 √ g(a, b; k, 0, 0) = ab.
(1.7)
Definition 1.2. [1] A convex function f (x), where is said to be Schur convex if for a ≥ b, then f (a) ≥ f (b), where a = (a1 , ..., an ), b = (b1 , ..., bn ) and x = (x1 , ..., xn ).
2
Main results
In this section, the convexity and Schur convexity of Gnan mean for two variables are established.
Lemma 2.1. Let a = (a1 , a2 ), b = (b1 , b2 ) and if a ≥ b, then G(a; k, α, β) ≥ G(b; k, α, β). Proof. For a ≥ b, then a1 ≥ b1 and a2 ≥ b2 . Also, for α, αβ > 0, then the following inequality holds h
α (k+1−i)aα 1 +ia2 k+1
i αβ
≥
h
α (k+1−i)bα 1 +ib2 k+1
i αβ
,
implies that,
αβ β1 α +iaα P (k+1−i)a k 1 1 2 k
i=1
k+1
≥
αβ β1 α +ibα P (k+1−i)b k 1 1 2 k
i=1
k+1
,
G(a, k, α, β) ≥ G(b, k, α, β). Thus the proof of Lemma 2.1 completes.
Theorem 2.1. The Gnan mean G(a, b; k, α, β) is convex if α,
39
β α
≥ 1 and concave if α,
β α
≤ 1.
Proof. From Definition 1.1 , let "
k
1X Gβ (a, b; k, α, β) = k i=1
(k + 1 − i)aα + ibα k+1
αβ # (2.1)
with simple manipulation eqn (2.1) rewritten as, β
kGβ (a, b; k, α, β) =
k X (k + 1 − i)aα + ibα α
(2.2)
k+1
i=1
put a = t and b = 1, then eqn (2.2) takes the form β
β
kG (t, 1; k, α, β) =
k X (k + 1 − i)tα + i1α α
(2.3)
k+1
i=1
differentiating both sides of the eqn (2.3) with respect to t twice, we have β −2 2 k X ∂2 β k + 1 − i α−1 (k + 1 − i)tα + i1α α k 2 G (t, 1; k, α, β) = t β(β − α) ∂t k+1 k+1 i=1 (2.4) β −1 k X (k + 1 − i)tα + i1α α k + 1 − i α−2 t + k+1 k+1 i=1 =β
k X i=1
then if
(k + 1 − i)t2α−2 k + 1 − i α−2 + (α − 1) )t (β − α) ((k + 1 − i)tα + i)(k + 1) k+1
∂2 ∂t2
(k + 1 − i)aα + ibα k+1
β G (t, 1; k, α, β) ≥ 0,
(α − 1) ≥ 0 and (β − α) ≥ 0.
Hence the proof of Theorem 2.1.
Theorem 2.2. The Gnan mean G(a, b; k, α, β) is Schur convex if α,
β α
≥ 1.
Proof. Consider the Gnan mean G(a1 , a2 ; k, α, β). Let a1 = λb1 + (1 − λ)b2 and a2 = (1 − λ)b1 + λb2 , leads to G(a1 , a2 ; k, α, β) = G[λb1 + (1 − λ)b2 , (1 − λ)b1 + λb2 ; k, α, β] = G[λ(b1 , b2 ) + (1 − λ)(b2 , b1 ); k, α, β] since G(a1 , a2 ; k, α, β) is convex, then
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αβ −1 ,
≤ λG(b1 , b2 ; k, α, β) + (1 − λ)G(b2 , b1 ; k, α, β) since G(a1 , a2 ; k, α, β) is symmetric, then = λG(b1 , b2 ; k, α, β) + (1 − λ)G(b1 , b2 ; k, α, β) = G(b1 , b2 ; k, α, β) This proves the Schur convexity of Gnan mean G(a, b; k, α, β) for two variables.
Note: Similar arguments holds good for the Gnan mean in dual forms.
Acknowledgements The authors are thankful to Prof. Zhi-Hua-Zhang for his valuable suggestions.
References [1] Bullen, P. S. Handbook of Means and Their Inequalities, Kluwer Acad. Publ., Dordrecht, 2003. [2] Lokesha, V., Zh.-H. Zhang, K. M. Nagaraja, Gnan Mean for two variables, Far East Journal of Mathematics, Vol. 31, 2008, No. 2, 263–272. [3] Lokesha, V., Zh.-H. Zhang, Y.-D. Wu, Two weighted product type means and its monotonicities, RGMIA Research Report Collection, Vol. 8, 2005, No. 1, Article 17. http://rgmia.vu.edu.au/v8n1.html [4] Webster, R. Convexity. Oxford University Press, Oxford, New York, Tokyo, 1994. [5] Zhang, Zh.-H., Y.-D. Wu, The generalized Heron mean and its dual form. Appl. Math. ENotes, Vol. 5, 2005, 16–23. http://www.math.nthu.edu.tw/˜amen/ [6] Xiao, Zh.-G., Zh.-H. Zhang. The Inequalities G 6 L 6 I 6 A in n Variables, J. Ineq. Pure & Appl. Math., Vol. 4, 2003, No. 2, Article 39. http://jipam.vu.edu.au/v4n2/110_02.pdf [7] Xiao, Zh.-G., Zh.-H. Zhang, V. Lokesha. The weighted Heron mean of several positive numbers, RGMIA Research Report Collection, Vol. 8, 2005, No. 3, Article 6. http://rgmia.vu.edu.au/v8n3.html [8] Xiao, Zh.-G., V. Lokesha, Zh.-H. Zhang. The weighted Heron dual mean of several positive numbers, RGMIA Research Report Collection, Vol. 8, 2005, No. 4, Article 19. http://rgmia.vu.edu.au/v8n4.html
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