CC BY
43Probl. Anal. Issues Anal. Vol. 4(22), No. 1, 2015, pp. 3-10 3
DOI: 10.15393/j3.art.2015.2709
UDC 517.18, 517.38
B. A. Bhayo, L. Yin
ON THE GENERALIZED CONVEXITY AND CONCAVITY
Abstract. A function f : R+ ^ R+ is (mi , m2)-convex (concave) if f (mi(x,y)) < (>) m2(f (x),f (y)) for all x,y G R+ = = (0, œ) and mi and m2 are two mean functions. Anderson et al. [1] studies the dependence of (mi, m2)-convexity (concavity) on mi and m,2 and gave the sufficient conditions of (mi, m2)-convexity and concavity of a function defined by Maclaurin series. In this paper, we make a contribution to the topic and study the (mi, m2)-convexity and concavity of a function where mi and m2 are identric and Alzer mean. As well, we prove a conjecture posed by Bruce Ebanks in [2].
Key words: logarithmic mean, identric mean, power mean, Alzer 'mean, convexity and concavity property, Ebanks' conjecture
2010 Mathematical Subject Classification: 33B10, 26D15, 26D99
1. Introduction. A function M : (0, to) x (0, to) ^ (0, to) is called a Mean function if
1) M(x,y)= M(y,x),
2) M(x, x) = x,
3) x < M(x,y) < y, whenever x < y,
4) M(ax, ay) = aM(x,y) for all a > 0,
Some examples of mean functions of two distinct positive real numbers are given below:
Arithmetic mean: A = A(x,y) = x + y, Geometric mean: G = G(x,y) = -\J~xy,
©Petrozavodsk State University, 2015
[MglHl
Harmonic mean: Logarithmic mean:
Identric mean: Alzer mean:
H = H (x,y) = L = L(x,y) =
1
I = I(x,y) = - —
K ' e\yy
A(1/x, 1/y)'
x - y
log(x) - log(y),
1 ( xx x 1/(x-y)
Jp Jp(x,y)
Power mean: Mt = Mt(x,y) =
p xp+1 — yp+1 p +1 xp — yp '
( fx1 + yl\ 1/t
(
2
^/xy,
P = 0, —1,
t = 0, t = 0.
It is easy to observe that J1 (x, y) = A(x, y), Jo (x, y) = L(x, y), J-2(x, y) = = H(x,y). For the historical background of these means we refer the reader to see [3]-[7] and the bibliography of these papers.
Before we introduce the earlier results from the literature we recall the following definition, see [1, 8].
Definition 1. Let f : I0 ^ (0, to) be continuous, where I0 is a sub-interval of (0, to). Let M and N be two any mean functions. We say that the function f is MN-convex (concave) if
f(M(x,y)) < (>)N(f(x),f(y)) for all x,y e Io .
Throughout the paper, the notion I0 is reserved for the sub-internal of (0, to).
In [1], Anderson, Vamanamurthy and Vuorinen studied the convexity and concavity of a function f with respect two mean values, and gave the following detailed result:
Lemma 1. [1, Theorem 2.4] Let f : I0 ^ (0, to) be a differentiable function. In items (4)-(9), let I0 = (0, 6), 0 < b < to. Then
1) f is AA-convex (concave) if and only if f '(x) is increasing (decreasing),
2) f is AG-convex (concave) if and only if f'(x)/f (x) is increasing (decreasing),
3) f is AH-convex (concave) if and only if f' (x)/f (x)2 is increasing (decreasing),
4) f is GA-convex (concave) if and only if xf' (x) is increasing (decreasing),
5) f is GG-convex (concave) if and only if xf '(x)/f (x) is increasing (decreasing),
6) f is GH-convex (concave) if and only if xf' (x)/f (x)2 is increasing (decreasing),
7) f is HA-convex (concave) if and only if x2f '(x) is increasing (decreasing),
8) f is HG-convex (concave) if and only if x2f' (x)/f (x) is increasing (decreasing),
9) f is HH-convex (concave) if and only if x2f '(x)/f (x)2 is increasing (decreasing).
After the publication ([1]), many authors have studied generalized convexity. For a partial survey of the recent results, see [9]. In [10], the following inequalities were studied:
Lemma 2. Let f : I0 ^ (0, to) be a continuous function, then
1) f is LL-convex (concave) if f is increasing and log-convex (concave),
2) f is AL-convex (concave) if f is increasing and log-convex (concave).
Recently, Baricz [11] took one step further and studied the MN-convexity(concavity) of a function f in a generalized way, and gave the following result:
Lemma 3. [11, Lemma 3] Let p, q e R and let f: [a, b] ^ (0, to) be a differentiable function for a, b e (0, to). The function f is (p, q)-convex ((p, q)-concave) if and only if x ^ x1-pf'(x)(f (x))q-1 is an increasing (decreasing) function.
It can be observed easily that (1,1)-convexity means the AA-convexity, (1, 0)-convexity means the AG-convexity, and (0, 0)-convexity means GG-convexity.
Lemma 4. [11, Theorem 7] Let a, b e (0, to) and f: [a, b] ^ (0, to) be a differentiable function. Denote g(x) = JX f (t) dt and h(x) = fX f (t) dt. Then
(a) If the function x ^ x1-pf (x) is increasing (decreasing), then g is (p, q)-convex (h is (p, q)-convex) for all p e R and q > 1.
(b) If the function x ^ x1 pf (x) is increasing (decreasing), then g is (p, q)-convex (h is (p, q)--convex) for all p = (0,1) and q < 0.
2. Main results. In this paper we make a contribution to the subject by giving the following theorems, which could be natural questions to ask after reading the above literature. These results are the extension of [1, 11, 10].
Theorem 1. Let f : I0 ^ (0, to) be a continuously differentiable, increasing and log-convex (concave) function. Then
I(f (x),f(y)) > (<)f (I(x,y)).
Theorem 2. Let f be a continuous real-valued function on (0, to). If f is strictly increasing and convex, then
Pf (x,y) < Rf (x,y) (1)
where
Pf(x,y) = f (W4(1/2)
and
1 [y
Rf (x,y) = f (t)dt.
y x J x
Remark 1. In [2], Ebanks defined Pf (x,y) and Rf (x,y), and proposed a problem for a continuous and strictly monotonic real-valued function f on (0, to) as follows:
Problem. Does strictly increasing and a convexity of f (or f" > 0) imply that Pf < Rf ?
It is obvious that the Theorem 2 gives an affirmative answer to the Ebanks' question.
Theorem 3. Let f : I0 ^ (0, to).
(1) If f (x) is continuously differentiable, strictly increasing(decreasing) and convex (concave) and f p-1(x)f' (x) is increasing on (0,1), then
JP(f (x),f (y)) > f (Jp(x,y)) Jp(f (x),f (y)) < f (A(x,y))
for p < 1 .
(2) If f (x) is continuously differentiable, strictly decreasing(increasing) and convex(concave) and fp-1 (x)f (x) is decreasing on (0,1), then
Jp(f (x),f (y)) > f (Jp(x,y)) Jp(f (x),f (y)) < f (A(x,y))
for p > 1.
3. Lemmas and proofs. We recall the following lemmas which will be used in the proofs of the theorems.
Lemma 5. [12] Let f, g : [a, 6] ^ R be integrable functions, both increasing or both decreasing. Furthermore, let p : [a, 6] ^ R be a positive, integrable function. Then
i-b i-b i-b i-b
/ p(x)f (x)dx^ / p(x)g(x)dx < / p(x)dx • / p(x)f (x)g(x)dx. (2)
J a J a J a J a
If one of the functions f or g is non-increasing and the other non-decreasing, then the inequality (2) is reversed.
Lemma 6. [13] If f (x) is a continuous and convex function on [a, 6], and ^(x) is continuous on [a, 6], then
I *(x)dx) < £f (^(x)) (3)
If function f (x) is continuous and concave on [a, 6], the inequality (3) is reversed.
Lemma 7. [4] Fix two positive number a, 6. Then
L(a,6) < I(a, 6) < A(a,6).
Lemma 8. [13] The function p ^ Jp (x,y) is strictly increasing on R \ \{0,-1}.
Proof of Theorem 1. Since the proof of part (2) is similar to part (1), we only prove the part (1) here. Clearly
lnI(f (x),f (y)) = f (x)lnf (x) - f ((y)ln f(y) - 1.
An easy computation and substitution t = f (u) yield
1 „„ , „ „ //((yx)) lntdt /yx lnf (u)f-(u)du
lnI(f(x), f(y)) = f(y; ,-= y „x——-. (4)
(f ( ),f (y)) fy) 1 dt Jyx f -(u)du ()
Since the functions f (x) and f(x) are increasing on I C (0, to) then, using Lemma 5 and assuming x > y, we have
x x x x
/ 1du - / ln f (u)f(u)du > / f(u)du - / ln f (u)du. (5)
y y y y
Combining (4) and (5), we obtain
fx ln f (u)du
ln I(f (x),f (y)) > Jy ' V ; , (6)
y - x
where we assume that x > y. Using the inequality (6), Lemmas 6 and 7, and considering the log-convexity of the function f (x), we get
I (f (x), f (y)) > ln f (y^ = ln f (^y^) > ln f (I (x, y)) .
This completes the proof. □
Proof of Theorem 2. Since f is a strictly increasing and convex function, then from Lemma 5 and the inequality G(x,y) < A(x,y) we obtain
D , , . Zxy f (u)du . , //xy udu'
Rf (x,y) > —-> f ^-
y - x y - x
^ ) > f ((xy)1/4 (^ f
= Pf (x,y).
This completes the proof. □
Proof of Theorem 3. For the proof of part (1), letting t = f (u), we get
ft tpdt /yx fP(u)f(u)du Jp(f (x),f(y)) = f tP-idt = iffP-1(u)f'(u)du.
By using Lemma 5, we obtain
Jp(f (x),f(y)) >
j'X f y - x '
Considering convexity of the function f (x) and using Lemmas 6 and 8, we get
Jp(f (x),f (y)) > = f(^) > f (Jp (x,y)),
which implies (1). The proof of part (2) follows similarly. □
The convexity and concavity properties of a real-valued function were studied in [1, 11, 10, 14] in the sense of many classical means, i.e. arithmetic mean, geometric mean, logarithmic mean, harmonic mean etc. In this paper, we made a contribution to the topic, and studied the convexity and concavity properties of a real-valued function with respect to identric mean, Alzer mean, as well as proved the conjecture posed by Ebanks.
Acknowledgment. The second author was supported by the National Natural Science Foundation of China 11401041, and by the Science Foundation of Binzhou University under grant BZXYL1303. Authors are indebted to an anonymous reviewer for giving insightful comments and providing the directions of improving the paper.
References
[1] Anderson G. D., Vamanamurthy M. K. and Vuorinen M. Generalized convexity and inequalities. J. Math. Anal. Appl., 2007, vol. 335, pp. 12941308.
[2] Ebanks B. Looking for a few good mean. Amer. Math. Monthly, 2012, vol. 119, no. 8, pp. 658-669.
[3] Alzer H. Two inequalities for means. C. R. Math. Rep. Acad. Sci. Canada, 1987, vol. 9, pp. 11-16.
[4] Alzer H. and Qiu S. L. Inequalities for means in two variables. Arch. Math., 2003, vol. 80, pp. 201-205.
[5] Carlson B. C. The logarithmic mean. Amer. Math. Monthly, 1972, vol. 79, pp. 615-618.
[6] Mitrinovic D. S. Analytic Inequalities. Springer, New York, USA, 1970.
[7] Sandor J. On the identric and logarithmic means. Aequat. Math., 1990, vol. 40, pp. 261-270.
[8] Aumann G. Konvexe Funktionen und Induktion bei Ungleichungen zwischen Mittelverten, Bayer. Akad. Wiss. Math.-Natur. Kl. Abh., Math. Ann., 1933, vol. 109, pp. 405-413.
[9] Anderson G. D., Vuorinen M. and Zhang X. Topics in special functions III. http://arxiv.org/abs/1209.1696.
[10] Bhayo B. A. and Yin L. Logarithmic mean inequality for generalized trigonometric and hyperbolic functions. Acta Univ. Sapientiae, Mathe-matica, 2014, vol. 6, no. 2, pp. 135-145.
[11] Baricz A. Geometrically concave univariate distributions. J. Math. Anal., 2010, 308 Appl., vol. 363(1), pp. 182-196.
[12] Qi F. and Huang Z. Inequalities of the complete elliptic integrals. Tamkang J. Math., 1998, vol. 29, no. 3, pp. 165-169.
[13] Kuang J. C. Applied inequalities (Second edition). Shan Dong Science and Technology Press. Jinan, 2002.
[14] Abramowitz M., Stegun I., eds. Handbook of mathematical functions with formulas, graphs and mathematical tables. National Bureau of Standards, Dover, New York, 1965.
Received December 21, 2014-In revised form, June 21, 2015.
Koulutuskeskus Salpaus (Salpaus Further Education) 7 Paasikivenkatu, FI-15110 Lahti, Finland E-mail: [email protected]
Binzhou University
Binzhou City, Shandong Province, 256603, China E-mail: [email protected]
