Volume 2010, Article ID 845390,8pages doi:10.1155/2010/845390
Research Article
Multiplicative Concavity of the Integral of
Multiplicatively Concave Functions
Yu-Ming Chu
1and Xiao-Ming Zhang
21Department of Mathematics, Huzhou Teachers College, Huzhou, Zhejiang 313000, China 2Haining College, Zhejiang TV University, Haining, Zhejiang 314400, China
Correspondence should be addressed to Yu-Ming Chu,[email protected]
Received 25 March 2010; Accepted 7 June 2010
Academic Editor: S. S. Dragomir
Copyrightq2010 Y.-M. Chu and X.-M. Zhang. 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 prove thatGx, y |xyftdt|is multiplicatively concave ona, b×a, biff : a, b ⊂
0,∞ → 0,∞is continuous and multiplicatively concave.
1. Introduction
For convenience of the readers, we first recall some definitions and notations as follows.
Definition 1.1. LetI⊆Rbe an interval. A real-valued functionf:I → Ris said to be convex if
f xy
2
≤ fx f
y
2 1.1
for allx, y∈I. Andfis called concave if−fis convex.
Definition 1.2. LetI ⊆0,∞be an interval. A real-valued functionf :I → 0,∞is said to be multiplicatively convex if
fxy≤
fxfy 1.2
Forx x1, x2∈R2{x1, x2:x1>0, x2>0}andα≥0, we denote
logxlogx1,logx2
,
xαxα1, xα2, 1.3
ex ex1, ex2. 1.4
Forx x1, x2,y y1, y2∈R2, we denote
xyx1y1, x2y2
. 1.5
Definition 1.3. A setE1⊆R2is said to be convex ifxy/2∈E1wheneverx, y∈E1. And a
setE2⊆R2is said to be multiplicatively convex ifx1/2y1/2∈E2wheneverx, y∈E2.
FromDefinition 1.3we clearly see thatE1 ⊆R2 is a multiplicatively convex set if and
only if logE1 {logx : x ∈ E1}is a convex set, andE2 ⊆ R2is a convex set if and only if
eE2 {ex:x∈E
2}is a multiplicatively convex set.
Definition 1.4. LetE ⊆ R2 be a convex set. A real-valued functionf : E → Ris said to be
convex if
f xy
2
≤ fx f
y
2 1.6
for allx, y∈E. Andfis said to be concave if−fis convex.
Definition 1.5. LetE ⊆ R2
be a multiplicatively convex set. A real-valued functionf : E →
0,∞is said to be multiplicatively convex if
f x1/2y1/2≤f1/2xf1/2y 1.7
for allx, y∈E. Andfis called multiplicatively concave if 1/fis multiplicatively convex.
From Definitions1.1and1.2, the following Theorem A is obvious.
Theorem A. Suppose thatIis a subinterval of0,∞andf:I → 0,∞is multiplicatively convex. Then
Fx log◦f◦exp : logI−→R 1.8
is convex. Conversely, ifJis an interval andF:J → Ris convex, then
fexp◦F◦log : expJ−→0,∞ 1.9
Equivalently, f is a multiplicatively convex function if and only if logfx is a convex function of logx. Modulo this characterization, the class of all multiplicatively convex functions was first considered by Motel1, in a beautiful paper discussing the analogues of the notion of convex function innvariables. However, the roots of the research in this area can be traced long before him. In a long time, the subject of multiplicative convexity seems to be even forgotten, which is a pity because of its richness. Recently, the multiplicative convexity has been the subject of intensive research. In particular, many remarkable inequalities were found via the approach of multiplicative convexitysee2–18.
The main purpose of this paper is to proveTheorem 1.6.
Theorem 1.6. Iff : a, b ⊂ 0,∞ → 0,∞is continuous and multiplicatively concave, then
Gx, y |xyftdt|is multiplicatively concave ona, b×a, b.
2. Lemmas and the Proof of
Theorem 1.6
For the sake of readability, we first introduce and establish several lemmas which will be used to predigest the proof ofTheorem 1.6.
Lemma 2.1can be derived from Definitions1.4and1.5.
Lemma 2.1. IfE1 ⊂ R2 is a multiplicatively convex set, andf : E1 → 0,∞is multiplicatively
convex (or concave, resp.), thenFx logfexis convex (or concave, resp.) onlogE
1 {logx:
x∈E1}. Conversely, ifE2 ⊂R2is a convex set, andF :E2 → Ris convex (or concave, resp.), then
fx eFlogxis multiplicatively convex (or concave, resp.) oneE2{ex:x∈E 2}.
Lemma 2.2see19. IfE⊂R2is a convex set, andf:E → Ris second-order differentiable, then
fis convex (or concave, resp.) if and only ifLxis a positive (or negative, resp.) semidefinite matrix for allx x1, x2∈E. Here
Lx
f11 f12 f21 f22
, 2.1
andfij ∂2fx
1, x2/∂xi∂xj,i, j1,2.
Making use of Lemmas2.1and2.2we get the followingLemma 2.3.
Lemma 2.3. If E ⊂ R2
is a multiplicatively convex set, and f : E → 0,∞ is second-order
differentiable, thenf is multiplicatively convex (or concave, resp.) if and only if Jxis a positive (or negative, resp.) semidefinite matrix for allx x1, x2∈E. Here
Jx ⎛ ⎜ ⎜ ⎝
ff11 f x1
f1−f2
1 ff12 −f1f2
ff21 −f1f2 ff22 f x2
f2−f22 ⎞ ⎟ ⎟
⎠, 2.2
fij ∂fx1, x2/∂xi∂xj, andfi∂fx1, x2/∂xi, i, j 1,2.
resp.) onI. If moreoverf is second-order differentiable, thenfis multiplicaively convex (or concave, resp.) if and only if
xfxfx−f2xfxfx≥or≤,resp.0 2.3
for allx∈I.
Lemma 2.5. Suppose that f : a, b ⊂ 0,∞ → 0,∞ is a second-order differentiable multiplicatively concave function. If gx axftdt, then g is also multiplicatively concave on
a, b.
Proof. Forx∈a, b, from the expression ofgxwe get
xgxgx−g2xgxgx xfx fx
x
a
ftdt−xf2x. 2.4
According toLemma 2.4, to prove thatgxis multiplicatively concave ona, b, it is sufficient to prove that
xfx fx
x
a
ftdt−xf2x≤0 2.5
for allx∈a, b. Next, set
Ex∈a, b:xfx fx≤0
x∈a, b:xf
x
fx ≤ −1
. 2.6
FromLemma 2.4we know thatxfx/fxis decreasing; the following three cases will complete the proof of inequality2.5.
Case 1. a∈E.ThenE a, b, andxfx fx≤0 for allx∈a, b; hence2.5is true for allx∈a, b.
Case 2. b /∈E. ThenEφ, that is,xfx fx>0 for allx∈a, b. Let
hx x
a
ftdt− xf
2x
Then from the multiplicative concavity offwe clearly see that
hx xfx
xfxfx−f2xfxfx
xfx fx2 ≤0
2.8
for allx∈a, b.
From2.7and2.8we get
hx≤ha − af
2a
afa fa ≤0 2.9
for allx∈a, b. Therefore, inequality2.5follows from2.7and2.9.
Case 3. a /∈E and b ∈ E. Then there exists a unique x0 ∈ a, bsuch that E x0, band
xfx fx> 0 forx∈a, x0. Making use of the similar argument as in Case2we know
that inequality2.5holds forx∈a, x0; this result andE x0, bimply that2.5holds for
allx∈a, b.
Lemma 2.6. Iff:a, b⊂0,∞ → 0,∞is a second-order differentiable multiplicatively concave function, then
fa afafb bfb
b
a
ftdt≤bf2bfa afa−af2afb bfb.
2.10
Proof. We divide the proof into five cases.
Case 1. fa afa 0. Then fromLemma 2.4we know thatxfx/fxis decreasing on a, b; hence we getfb bfb≤0.It is obvious that inequality2.10holds in this case.
Case 2. fb bfb 0.Then2.10follows fromfa afa≥0.
Case 3. fa afa<0.Thenfx xfx<0 for allx∈a, b. From2.7and2.8we get
hb b
a
ftdt− bf
2b
bfb fb ≤ −
af2a
afa fa ha. 2.11
Therefore, inequality2.10follows from inequality2.11andfx xfx<0.
Case 4. fb bfb > 0.Thenfx xfx > 0 for allx ∈ a, b; hence inequality2.10
follows from2.11andfx xfx>0.
Case 5. fa afa>0, fb bfb<0. Then we clearly see that2.10is true.
Proof. Forx, y∈a, b×a, b, without loss of generality, we assume thaty≤x. Then simple computations lead to
GG11G xG
1−G
2
1 fx
x
y
ftdtfx x
x
y
ftdt−f2x, 2.12
GG22G yG
2−G
2
2 −f
y
x
y
ftdt−f
y y
x
y
ftdt−f2y, 2.13
GG12−G1G2GG21−G1G2fxfy. 2.14
From Lemma 2.5 we know that Fx yxftdt is multiplicatively concave; then
Lemma 2.4leads to
xFxFx−F2xFxFx xfx fx
x
y
ftdt−xf2x≤0. 2.15
Combining2.12and2.15we get
GG11G xG
1−G
2
1 ≤0. 2.16
Equations2.12–2.14andLemma 2.6yield
GG11G xG
1−G
2
1
GG22G yG
2−G
2
2
−GG12−G1G2×GG21−G2G1
x
yftdt
xy
xf2xfyyfy−yf2yfx xfx
−fx xfxfyyfy
x
y
ftdt
≥0.
2.17
Therefore,Lemma 2.7follows from2.16and2.17together withLemma 2.3.
Lemma 2.8see20. For each continuous convex functionf:a, b → R, there exists a sequence of infinitely differentiable convex functionsfn:a, b → R, n1,2,3, . . ., such that{fn}converges
uniformly tofona, b.
From Definitions 1.1 and 1.2, Theorem A, and Lemma 2.8 we can get Lemma 2.9
immediately.
Lemma 2.9. For each continuous multiplicatively convex (or concave, resp.) functionf : a, b ⊆ 0,∞ → 0,∞, there exists a sequence of infinitely differentiable multiplicatively convex (or concave, resp.) functionsfn :a, b → 0,∞, n 1,2,3, . . ., such that{fn}converges uniformly
Proof ofTheorem 1.6. Since f : a, b ⊆ 0,∞ → 0,∞ is a continuous multiplicatively concave function, from Lemma 2.9 we know that there exists a sequence of infinitely differentiable multiplicatively concave function fn : a, b → 0,∞, n 1,2,3, . . ., such
that{fn}converges uniformly tofona, b.
For x, y ∈ a, b× a, b, taking Gnx, y |
y
xfntdt|, n 1,2,3, . . ., then by Lemma 2.7we clearly see thatGnx, yis multiplicatively concave ona, b×a, band
lim
n→ ∞Gn
x, y y
x
ftdtGx, y. 2.18
Therefore,Theorem 1.6follows fromDefinition 1.5and2.18.
Acknowledgments
The research was supported by the Natural Science Foundation of China under Grant 60850005 and the Innovation Team Foundation of the Department of Education of Zhejiang Province under Grant T200924.
References
1 P. Montel, “Sur les functions convexes et les fonctions sousharmoniques,”Journal de Math´ematiques, vol. 7, no. 9, pp. 29–60, 1928.
2 C. P. Niculescu, “Convexity according to the geometric mean,”Mathematical Inequalities & Applications, vol. 3, no. 2, pp. 155–167, 2000.
3 C. P. Niculescu, “Convexity according to means,”Mathematical Inequalities & Applications, vol. 6, no. 4, pp. 551–579, 2003.
4 C. P. Niculescu and L.-E. Persson, Convex Functions and Their Applications, CMS Books in Mathematics/Ouvrages de Math´ematiques de la SMC, 23, Springer, New York, NY, USA, 2006. 5 G. D. Anderson, M. K. Vamanamurthy, and M. Vuorinen, “Generalized convexity and inequalities,”
Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1294–1308, 2007.
6 C. E. Finol and M. W ´ojtowicz, “Multiplicative properties of real functions with applications to classical functions,”Aequationes Mathematicae, vol. 59, no. 1-2, pp. 134–149, 2000.
7 T. Trif, “Convexity of the gamma function with respect to H ¨older means,” inInequality Theory and Applications. Vol. 3, pp. 189–195, Nova Science Publishers, Hauppauge, NY, USA, 2003.
8 K. Z. Guan, “A class of symmetric functions for multiplicatively convex function,”Mathematical Inequalities & Applications, vol. 10, no. 4, pp. 745–753, 2007.
9 X.-M. Zhang and Z.-H. Yang, “Differential criterion of n-dimensional geometrically convex functions,”Journal of Applied Analysis, vol. 13, no. 2, pp. 197–208, 2007.
10 Y. Chu, X. M. Zhang, and G.-D. Wang, “The Schur geometrical convexity of the extended mean values,”Journal of Convex Analysis, vol. 15, no. 4, pp. 707–718, 2008.
11 Y. M. Chu, W. F. Xia, and T. H. Zhao, “Schur convexity for a class of symmetric functions,”Science China Mathematics, vol. 53, no. 2, pp. 465–474, 2010.
12 W.-F. Xia and Y.-M. Chu, “Schur-convexity for a class of symmetric functions and its applications,” Journal of Inequalities and Applications, vol. 2009, Article ID 493759, 15 pages, 2009.
13 X.-M. Zhang and Y.-M. Chu, “A double inequality for gamma function,”Journal of Inequalities and Applications, vol. 2009, Article ID 503782, 7 pages, 2009.
14 Y.-M. Chu and Y.-P. Lv, “The Schur harmonic convexity of the Hamy symmetric function and its applications,”Journal of Inequalities and Applications, vol. 2009, Article ID 838529, 10 pages, 2009. 15 T.-H. Zhao, Y.-M. Chu, and Y.-P. Jiang, “Monotonic and logarithmically convex properties of a
16 X.-H. Zhang, G.-D. Wang, and Y.-M. Chu, “Convexity with respect to H ¨older mean involving zero-balanced hypergeometric functions,”Journal of Mathematical Analysis and Applications, vol. 353, no. 1, pp. 256–259, 2009.
17 X.-M. Zhang and Y.-M. Chu, “The geometrical convexity and concavity of integral for convex and concave functions,”International Journal of Modern Mathematics, vol. 3, no. 3, pp. 345–350, 2008. 18 X.-M. Zhang and Y.-M. Chu, “A new method to study analytic inequalities,”Journal of Inequalities and
Applications, vol. 2010, Article ID 698012, 13 pages, 2010.
19 A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications, vol. 143 of Mathematics in Science and Engineering, Academic Press, New York, NY, USA, 1979.