Page 11 - REVISTA MATINF Nr. 5

O inegalitate pentru funct¸ii convexe/concave 11

Propozit , ia 7. ([7]) Dac˘a a 1 , a 2 , . . . , a n > 0 cu notat ,iile a 1 +a 2 +. . .+a n = S s , i a 1 ·a 2 ·. . .·a n = P,
avem:

€ Š S
a 2
a 1
a · a · . . . · a a n · (S−a 1 ) S−a 1 · (S−a 2 ) S−a 2 · . . . · (S−a n ) S−a n ≥ (n−1) n−1 · P . (12)
1 2 n
Demonstrat¸ie. Consider˘am funct , ia f : (0, ∞) → R, f(x) = x ln x, care este funct , ie strict
convex˘a. Calcul˘am s , i aici:
n  ‹ n n
X X X
S − a k S − a k S − a k S − a k
a k f = a k · ln = (S − a k ) ln
a k a k a k a k
k=1 k=1 k=1
n  ‹ n  ‹
X S − a k S−a k Y S − a k S−a k
= ln = ln
a k a k
k=1 k=1
n n n
(S − a k ) S−a k 1 1
Y Y Y
a k
a k
= ln = ln n · a (S − a k ) S−a k = ln a (S − a k ) S−a k ,
k
k
a S−a k Q S P S
k=1 k a k=1 k=1
k
k=1
(n−1)S
respectiv f (n−1) · S= [(n−1) ln (n−1)] · S=ln (n−1) .
Aplicˆand inegalitatea (C) se obt , ine inegalitatea (12).
Egalitatea are loc dac˘a s , i numai dac˘a a 1 = a 2 = . . . = a n .
Bibliografie

[1] J.L.W.V. Jensen, Sur les fonctions convexes et les in´egalit´es entre les valeurs moyennes,
Acta Mathematica, 30 (1), pp. 175–193, 1906.

[2] D. M˘arghidanu, Generalizations and refinements for Bergstr¨om and Radon’s Inequalities,
Journal of Sciences and Arts, Year 8, No. 1 (8), 2008, on-line, http://www.icstm.ro/DOCS/
josa/josa_2008_1/cuprins.htm
[3] D. M˘arghidanu, Generaliz˘ari ale inegalit˘at ,ilor lui Young, H¨older, Rogers s , i Minkowski,
Gazeta Matematic˘a, seria A, Anul XXVI (CV), nr. 3 / 2008.

[4] D. M˘arghidanu, Proposed problem, Mathematical Inequalities, https://www.facebook.
com/photo.php?fbid=3378302362228848&set=gm.2648165092138202&type=3&theater&
ifg=1
[5] D. M˘arghidanu, Proposed problem, Romanian Mathematical Magazine, https:
//www.facebook.com/photo.php?fbid=2585007641829030&set=gm.1838155749642029&
type=3&theater
[6] D. M˘arghidanu, Proposed problem, Math Facts, https://www.facebook.com/photo.php?
fbid=3390923744300043&set=gm.739274503502386&type=3&theater&ifg=1
[7] D. M˘arghidanu, Proposed problem, Groupe Matheux, https://www.facebook.com/photo.
php?fbid=3400655483326869&set=gm.1593674704140714&type=3&theater&ifg=1
[8] D.S. Mitrinovi´c, J.E. Pecari´c, A.M. Fink, Classical and New Inequalities in Analysis, Kluwer
Acad. Press., 1993.

[9] P.M. Vasic, J.E. Pecaric, On the Jensen Inequality, Univ. Beograd. Publ. Electrotehn. Fak.,
Ser. Mat. Fiz., No. 634 – 677, pp. 50-54, 1979.