Jounal of Inequalities in Pue and Applied Mathematics COEFFICIENT INEQUALITY FOR A FUNCTION WHOSE DERIVATIVE HAS A POSITIVE REAL PART S. ABRAMOVICH, M. KLARIČIĆ BAKULA AND S. BANIĆ Depatment of Mathematics Univesity of Haifa Haifa, 31905, Isael. EMail: abamos@math.haifa.ac.il Depatment of Mathematics Faculty of Natual Sciences, Mathematics and Education Univesity of Split Teslina 12, 21000 Split Coatia. EMail: milica@pmfst.h Faculty of Civil Engineeing and Achitectue Univesity of Split Matice hvatske 15, 21000 Split Coatia. EMail: Senka.Banic@gadst.h volume 7, issue 2, aticle 70, 2006. Received 03 Januay, 2006; accepted 20 Januay, 2006. Communicated by: C.P. Niculescu Abstact Home Page c 2000 Victoia Univesity ISSN electonic: 1443-5756 002-06
Abstact A vaiant of Jensen-Steffensen s inequality is consideed fo convex and fo supequadatic functions. Consequently, inequalities fo powe means involving not only positive weights have been established. 2000 Mathematics Subject Classification: 26D15, 26A51. Key wods: Jensen-Steffensen s inequality, Monotonicity, Supequadaticity, Powe means. 1 Intoduction......................................... 3 2 Vaiants of fo Positive Supequadatic................................ 10 3 An Altenative Poof of Theoem B..................... 19 Refeences Page 2 of 27
1. Intoduction. Let I be an inteval in R and f : I R a convex function on I. If ξ = ξ 1,..., ξ m is any m-tuple in I m and p = p 1,..., p m any nonnegative m- tuple such that m p i > 0, then the well known Jensen s inequality see fo example [7, p. 43] 1.1 f 1 P m p i ξ i 1 P m p i f ξ i holds, whee P m = m p i. If f is stictly convex, then 1.1 is stict unless ξ i = c fo all i {j : p j > 0}. It is well known that the assumption p is a nonnegative m-tuple can be elaxed at the expense of moe estictions on the m-tuple ξ. If p is a eal m-tuple such that 1.2 0 P j P m, j = 1,..., m, P m > 0, whee P j = j p i, then fo any monotonic m-tuple ξ inceasing o deceasing in I m we get ξ = 1 p i ξ i I, P m and fo any function f convex on I 1.1 still holds. Inequality 1.1 consideed unde conditions 1.2 is known as the Jensen-Steffensen s inequality [7, p. 57] fo convex functions. In his pape [5] A. McD. Mece consideed some monotonicity popeties of powe means. He poved the following theoem: Page 3 of 27
Theoem A. Suppose that 0 < a < b and a x 1 x 2 x n b hold with at least one of the x k satisfying a < x k < b. If w = w 1,..., w n is a positive n-tuple with n w i = 1 and < < s < +, then a < Q a, b; x < Q s a, b; x < b, whee fo all eal t 0, and Q t a, b; x Q 0 a, b; x ab G, a t + b t w i x t i G = n x w i i. In his next pape [6], Mece gave a vaiant of Jensen s inequality fo which Witkowski pesented in [8] a shote poof. This is stated in the following theoem: Theoem B. If f is a convex function on an inteval containing an n-tuple x = x 1,..., x n such that 0 < x 1 x 2 x n and w = w 1,..., w n is a positive n-tuple with n w i = 1, then f x 1 + x n w i x i f x 1 + f x n w i f x i. This theoem is a special case of the following theoem poved in [4] by Abamovich, Klaičić Bakula, Matić and Pečaić: 1 t Page 4 of 27
Theoem C [4, Th. 2]. Let f : I R, whee I is an inteval in R and let [a, b] I, a < b. Let x = x 1,..., x n be a monotonic n-tuple in [a, b] n and v = v 1,..., v n a eal n tuple such that v i 0, i = 1,..., n, and 0 V j V n, j = 1,..., n, V n > 0, whee V j = j v i. If f is convex on I, then 1.3 f a + b 1 v i x i f a + f b 1 v i f x i. V n V n In case f is stictly convex, the equality holds in 1.3 iff one of the following two cases occus: 1. eithe x = a o x = b, 2. thee exists l {2,..., n 1} such that x = x 1 + x n x l and 1.4 x 1 = a, x n = b o x 1 = b, x n = a, V j x j 1 x j = 0, j = 2,..., l, V j x j x j+1 = 0, j = l,..., n 1, whee V j = n i=j v i, j = 1,..., n and x = 1/V n n v ix i. In the special case whee v > 0 and f is stictly convex, the equality holds in 1.4 iff x i = a, i = 1,..., n, o x i = b, i = 1,..., n. Hee, as in the est of the pape, when we say that an n-tuple ξ is inceasing deceasing we mean that ξ 1 ξ 2 ξ n ξ 1 ξ 2 ξ n. Similaly, Page 5 of 27
when we say that a function f : I R is inceasing deceasing on I we mean that fo all u, v I we have u < v fu fv u < v fu fv. In Section 2 we efine Theoems A, B, and C. These efinements ae achieved by supequadatic functions which wee intoduced in [1] and [2]. As Jensen s inequality fo convex functions is a genealization of Hölde s inequality fo f x = x p, p 1, so the inequalities satisfied by supequadatic functions ae genealizations of the inequalities satisfied by the supequadatic functions f x = x p, p 2 see [1], [2]. Fist we quote some definitions and state a list of basic popeties of supequadatic functions. Definition 1.1. A function f : [0, R is supequadatic povided that fo all x 0 thee exists a constant Cx R such that 1.5 f y f x f y x C x y x fo all y 0. Definition 1.2. A function f : [0, R is said to be stictly supequadatic if 1.5 is stict fo all x y whee xy 0. Lemma A [2, Lemma 2.3]. Suppose that f is supequadatic. Let ξ i 0, i = 1,..., m, and let ξ = m p iξ i, whee p i 0, i = 1,..., m, and m p i = 1. Then p i f ξ i f ξ p i f ξi ξ. Lemma B [1, Lemma 2.2]. Let f be supequadatic function with Cx as in Definition 1.1. Then: Page 6 of 27
i f0 0, ii if f0 = f 0 = 0 then Cx = f x wheneve f is diffeentiable at x > 0, iii if f 0, then f is convex and f0 = f 0 = 0. In [3] the following efinement of Jensen s Steffensen s type inequality fo nonnegative supequadatic functions was poved: Theoem D [3, Th. 1]. Let f : [0, [0, be a diffeentiable and supequadatic function, let ξ be a nonnegative monotonic m-tuple in R m and p a eal m-tuple, m 3, satisfying Let ξ be defined as Then 1.6 0 P j P m, j = 1,..., m, P m > 0. p i f ξ i P m f ξ ξ = 1 P m p i ξ i. k 1 P i f ξ i+1 ξ i + P k f ξ ξ k + P k+1 f ξ k+1 ξ + P i f ξ i ξ i 1 i=k+2 Page 7 of 27
[ P i + i=k+1 P i ] f m p i ξ ξi k P i + m m m 1 P m f p i ξ ξi, m 1 P m whee P i = m j=i p j and k {1,..., m 1} satisfies i=k+1 P i ξ k ξ ξ k+1. In case f is also stictly supequadatic, inequality p i f ξ i P m f ξ m > m 1 P m f p ξi i ξ m 1 P m holds fo ξ > 0 unless one of the following two cases occus: 1. eithe ξ = ξ 1 o ξ = ξ m, 2. thee exists k {3,..., m 2} such that ξ = ξ k and { Pj ξ j ξ j+1 = 0, j = 1,..., k 1 1.7 P j ξ j ξ j 1 = 0, j = k + 1,..., m. In these two cases p i f ξ i P m f ξ = 0. Page 8 of 27
In Section 2 we efine Theoem B and Theoem C fo functions which ae supequadatic and positive. One of the efinements is deived easily fom Theoem D. We use in Section 3 the following theoem [7, p. 323] to give an altenative poof of Theoem B. Theoem E. Let I be an inteval in R, and ξ, η two deceasing m-tuples such that ξ, η I m. Let p be a eal m-tuple such that 1.8 p i ξ i p i η i fo k = 1, 2,..., m 1, and 1.9 p i ξ i = p i η i. Then fo evey continuous convex function f : I R we have 1.10 p i f ξ i p i f η i. Page 9 of 27
2. Vaiants of fo Positive Supequadatic In this section we efine in two ways Theoem C fo functions which ae supequadatic and positive. The efinement in Theoem 2.1 follows by showing that it is a special case of Theoem D fo specific p. The efinement in Theoem 2.2 follows the steps in the poof of Theoem B given by Witkowski in [8]. Theefoe the second efinement is confined only to the specific p given in Theoem B, which means that what we get is a vaiant of Jensen s inequality and not of the moe geneal Jensen-Steffensen s inequality. Theoem 2.1. Let f : [0, [0, and let [a, b] [0,. Let x = x 1,..., x n be a monotonic n tuple in [a, b] n and v = v 1,..., v n a eal n- tuple such that v i 0, i = 1,..., n, 0 V j V n, j = 1,..., n, and V n > 0, whee V j = j v i. If f is diffeentiable and supequadatic, then 2.1 f a + f b 1 v i f x i f a + b 1 v i x i V n V n n + 1 f b a 1 n V n v i a + b x i 1 n V n j=1 v jx j. n + 1 In case f is also stictly supequadatic and a > 0, inequality 2.1 is stict unless one of the following two cases occus: 1. eithe x = a o x = b, Page 10 of 27
2. thee exists l {2,..., n 1} such that x = x 1 + x n x l and x 1 = a, x n = b o x 1 = b, x n = a 2.2 V j x j 1 x j = 0, j = 2,..., l, V j x j x j+1 = 0, j = l,..., n 1, whee V j = n i=j v i, j = 1,..., n, and x = 1 n V n v ix i. In these two cases we have f a + f b 1 v i f x i f a + b 1 v i x i = 0. V n V n In the special case whee v > 0 and f is also stictly supequadatic, the equality holds in 2.1 iff x i = a, i = 1,..., n, o x i = b, i = 1,..., n. Poof. Suppose that x is an inceasing n-tuple in [a, b] n. The poof of the theoem is an immediate esult of Theoem D, by defining the n + 2-tuples ξ and p as ξ 1 = a, ξ i+1 = x i, i = 1,..., n, ξ n+2 = b p 1 = 1, p i+1 = v i /V n, i = 1,..., n, p n+2 = 1. Then we get 2.1 fom the last inequality in 1.6 and fom the fact that in ou special case we have P i + n+2 i=k+1 P i n + 1, Page 11 of 27
fo P i = i j=1 p j and P i = n+2 j=i p j, and ξ = 1 P m p i ξ i = a + b 1 V n v i x i = a + b x. The poof of the equality case and the special case whee v > 0 follows also fom Theoem D. We have P j = V j V n, j = 1,..., n, P n+1 = 0, P n+2 = 1, P 1 = 1, P 2 = 0, P j = V j 2 V n, j = 3,..., n + 2. Obviously, ξ = ξ 1 is equivalent to x = b and ξ = ξ n+2 is equivalent to x = a. Also, the existence of some k {3,..., m 2} such that ξ = ξ k and that 1.7 holds is equivalent to the existence of some l {2,..., n 1} such that x = x 1 +x n x l = a+b x l and that 2.2 holds. Theefoe, applying Theoem D we get the desied conclusions. In the case when x is deceasing we simply eplace x and v with x = x n,..., x 1 and ṽ = v n,..., v 1, espectively, and then ague in the same manne. In the special case that v > 0 also, V i > 0 and V j > 0, i = 1,..., n, and theefoe accoding to 2.2 equality holds in 2.1 only when eithe x 1 = = x n = a o x 1 = = x n = b. In the following theoem we will pove a efinement of Theoem B. Without loss of geneality we assume that n v i = 1. Page 12 of 27
Theoem 2.2. Let f : [0, [0, and let [a, b] [0,, a < b. Let x = x 1,..., x n be an n-tuple in [a, b] n and v = v 1,..., v n a eal n-tuple such that v 0 and n v i = 1. If f is supequadatic we have f a + f b v i f x i f a + b v i x i 2.3 v i f v j x j x i j=1 [ xi a + 2 v i b a f b x i + b x ] i b a f x i a 2 xi a b x i v i f v j x j x i + 2 v i f. b a j=1 If f is stictly supequadatic and v > 0 equality holds in 2.3 iff x i = a, i = 1,..., n, o x i = b, i = 1,..., n. Poof. The poof follows the technique in [8] and efines the esult to positive supequadatic functions. Fom Lemma A we know that fo any λ [0, 1] the following holds: 2.4 λf a + 1 λ f b f λa + 1 λ b λf a λa 1 λ b + 1 λ f b λa 1 λ b = λf 1 λ a b + 1 λ f λ b a = λf 1 λ b a + 1 λ f λ b a. Page 13 of 27
Also, fo any x i [a, b] thee exists a unique λ i [0, 1] such that x i = λ i a + 1 λ i b. We have 2.5 f a+f b v i f x i = f a+f b v i f λ i a + 1 λ i b. Applying 2.4 on evey x i = λ i a + 1 λ i b in 2.5 we obtain f a + f b v i f x i 2.6 = f a + f b + [ v i λi f a 1 λ i f b + λ i f 1 λ i b a + 1 λ i f λ i b a ] v i [1 λ i f a + λ i f b] + v i [λ i f 1 λ i b a + 1 λ i f λ i b a]. Applying again 2.4 on 2.6 we get 2.7 f a + f b v i f x i v i f 1 λ i a + λ i b + 2 v i [λ i f 1 λ i b a + 1 λ i f λ i b a]. Page 14 of 27
Applying again Lemma A on 2.7 we obtain 2.8 f a + f b v i f x i f v i [1 λ i a + λ i b] = f + + 2 v i f 1 λ i a + λ i b v j [1 λ j a + λ j b] j=1 v i [1 λ i f λ i b a + λ i f 1 λ i b a] a + b v i x i + v i f v j x j x i j=1 [ xi a + 2 v i b a f b x i + b x ] i b a f x i a, and this is the fist inequality in 2.3. Since f is a nonnegative supequadatic function, fom Lemma B we know that it is also convex, so fom 2.8 we have [ xi a v i b a f b x i + b x ] i b a f x i a hence, the second inequality in 2.3 is poved. 2 b xi x i a v i f, b a Page 15 of 27
Fo the case when f is stictly supequadatic and v > 0 we may deduce that inequalities 2.6 and 2.7 become equalities iff each of the λ i, i = 1,..., n, is eithe equal to 1 o equal to 0, which means that x i {a, b}, i = 1,..., n. Howeve, since we also have v j x j x i = 0, i = 1,..., n, j=1 we deduce that x i = a, i = 1,..., n, o x i = b, i = 1,..., n. This completes the poof of the theoem. Coollay 2.3. Let v = v 1,..., v n be a eal n-tuple such that v 0, n v i = 1 and let x = x 1,..., x n be an n-tuple in [a, b] n, 0 < a < b. Then fo any eal numbes and s such that s 2 we have 2.9 s Qs a, b; x 1 Q a, b; x s 1 Q a, b; x s v i v j x j x i j=1 ] x +2 v i a i b a b x i s b x + i b a x i a s s 1 Q a, b; x s v i v j x j x i j=1 Page 16 of 27
+2 v i 2 x i a b x i b a ] s, whee Q p a, b; x = a p + b p n v ix i 1 p, p R \ {0}. If s > 2 and v > 0, the equalities hold in 2.9 iff x i = a, i = 1,..., n o x i = b, i = 1,..., n. Poof. We define a function f : 0, 0, as f x = x s. It can be easily checked that fo any eal numbes and s such that s 2 the function f is supequadatic. We define a new positive n-tuple ξ in [a, b ] as ξ i = x i, i = 1,..., n. Fom Theoem 2.2 we have s a s + b s v i x s i a + b v i x i 2.10 v i v j x j x i j=1 s [ x + 2 v i a i b a b x i s s v i v j x j x i + 2 j=1 v i 2 x i a b x i b a ] b x + i b a x i a s s 0. Page 17 of 27
We have a s + b s v i x s i a + b v i x i s = Q s a, b; x s Q a, b; x s so fom 2.10 the inequalities in 2.9 follow. The equality case follows fom the equality case in Theoem 2.2, as the function f x = x s is stictly supequadatic fo s > 2. Remak 1. It is an immediate esult of Coollay 2.3 that if s > 2 and thee is at least one j {1,..., n} such that then fo this j we have s Qs a, b; x 1 > Q a, b; x v j x j a b x j > 0, 2v j Q a, b; x s 2 x j a b x j b a s > 0. Page 18 of 27
3. An Altenative Poof of Theoem B In this section we give an inteesting altenative poof of Theoem B based on Theoem E. To cay out that poof we need the following technical lemma. Lemma 3.1. Let y = y 1,..., y m be a deceasing eal m-tuple and p = p 1,..., p m a nonnegative eal m-tuple with m p i = 1. We define y = p i y i and the m-tuple y = y, y,..., y. Then the m-tuples η = y, ξ = y and p satisfy conditions 1.8 and 1.9. Poof. Note that y is a convex combination of y 1, y 2,..., y m, so we know that y m y y 1. Fom the definitions of the m-tuples ξ and η we have p i ξ i = y p i = y = p i y i = p i η i. Hence, condition 1.9 is satisfied. Futhemoe, fo k = 1, 2,..., m 1 we Page 19 of 27
have p i η i p i ξ i = = p i y i y p i y i j=1 p i p j y j p i. Since m p i = 1, we can wite p i η i p i ξ i = p j + = = = j=1 j=k+1 p j p i p i p j j=k+1 p i y i m j=k+1 m j=k+1 p j y i p i j=k+1 p j y i y j. p i y i p j y j + m j=k+1 p j y j p j y j j=1 j=k+1 p j y j p i Page 20 of 27
Since p is nonnegative and y is deceasing, we obtain p i η i p i ξ i 0, k = 1, 2,..., m 1, which means that condition 1.8 is satisfied as well. Now we can give an altenative poof of Theoem B which is mainly based on Theoem E. Poof of Theoem B. Since x = n w ix i is a convex combination of x 1, x 2,..., x n it is clea that thee is an s {1, 2,..., n 1} such that that is, x 1 x s x x s+1 x n, 3.1 x 1 x s x x s+1 x n. Adding x 1 + x n to all the inequalities in 3.1 we obtain x n x 1 + x n x s x 1 + x n x x 1 + x n x s+1 x 1, which gives us 3.2 x 1 + x n x = x 1 + x n w i x i [x 1, x n ]. Page 21 of 27
We use 1.10 to pove the theoem. Fo this, we define the n + 2-tuples ξ, η and p as follows: η 1 = x n, η 2 = x n, η 3 = x n 1,..., η n = x 2, η n+1 = x 1, η n+2 = x 1, p 1 = 1, p 2 = w n, p 3 = w n 1,..., p n = w 2, p n+1 = w 1, p n+2 = 1, n+2 ξ 1 = ξ 2 = = ξ n+2 = η, η = p i η i = x 1 + x n w j x j. j=1 It is easily veified that ξ and η ae deceasing and that n+2 p i = 1. It emains to see that ξ, η and p satisfy conditions 1.8 and 1.9. Condition 1.9 is tivially fulfilled since n+2 n+2 n+2 p i ξ i = η p i = η = p i η i. Futhe, we have ξ i = η, i = 1, 2,..., n + 2. To pove 1.8, we need to demonstate that 3.3 η p i p i η i, k = 1, 2,..., n + 1. Fo k = 1, 3.3 becomes η x n, and this holds because of 3.2. On the othe hand, fo k = n + 1, 3.3 becomes η 1 w i x n w i x i, Page 22 of 27
that is, 0 x n x, and this holds because of 3.2. If k {2,..., n}, 3.3 can be ewitten and in its stead we have to pove that 3.4 η 1 w i x n w i x i. i=n+2 k Let us conside the deceasing n-tuple y, whee We have y = = i=n+2 k y i = x 1 + x n x i, i = 1, 2,..., n. w i y i w i x 1 + x n x i = x 1 + x n w i x i = x 1 + x n x = η. If we apply Lemma 3.1 to the n-tuple y and to the weights w, then m = n and fo all l {1, 2,..., n 1} the inequality y l w i l w i x 1 + x n x i Page 23 of 27
holds. Taking into consideation that y = η, l w i = 1 n i=l+1 w i and changing indices as l = n + 1 k, we deduce that 3.5 η 1 i=n+2 k w i n+1 k w i x 1 + x n x i, fo all k {2,..., n}. The diffeence between the ight side of 3.4 and the ight side of 3.5 is n+1 k x n w i x i w i x 1 + x n x i i=n+2 k = x n = x n 1 = x n = i=n+2 k n+1 k i=n+2 k i=n+2 k w i n+1 k w i x i x n w i i=n+2 k i=n+2 k n+1 k w i x n x i + w i x i w i w i x i n+1 k n+1 k n+1 k w i x 1 x i w i x 1 x i w i x i x 1 0, since w is nonnegative and x is inceasing. Theefoe, the inequality n+1 k 3.6 w i x 1 + x n x i x n w i x i i=n+2 k w i x 1 x i Page 24 of 27
holds fo all k {2,..., n}. Fom 3.5 and 3.6 we obtain 3.4. This completes the poof that the m-tuples ξ, η and p satisfy conditions 1.8 and 1.9 and we can apply Theoem E to obtain n+2 p i f η f x n w i f x i + f x 1. Taking into consideation that n+2 p i = 1 and η = x 1 + x n n j=1 w jx j we finally get f x 1 + x n w i x i f x 1 + f x n w i f x i. Page 25 of 27
Refeences [1] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Refining Jensen s inequality, Bull. Math. Soc. Math. Roum., 47 2004, 3 14. [2] S. ABRAMOVICH, G. JAMESON AND G. SINNAMON, Inequalities fo aveages of convex and supequadatic functions, J. Inequal. in Pue and Appl. Math., 54 2004, At. 91. [ONLINE: http://jipam.vu.edu. au/aticle.php?sid=444]. [3] S. ABRAMOVICH, S. BANIĆ, M. MATIĆ AND J. PEČARIĆ, Jensen- Steffensen s and elated inequalities fo supequadatic functions, submitted fo publication. [4] S. ABRAMOVICH, M. KLARIČIĆ BAKULA, M. MATIĆ AND J. PEČARIĆ, A vaiant of Jensen-Steffensen s inequality and quasi-aithmetic means, J. Math. Anal. Applics., 307 2005, 370 385. [5] A. McD. MERCER, A monotonicity popety of powe means, J. Inequal. in Pue and Appl. Math., 33 2002, At. 40. [ONLINE: http: //jipam.vu.edu.au/aticle.php?sid=192]. [6] A. McD. MERCER, A vaiant of Jensen s inequality, J. Inequal. in Pue and Appl. Math., 44 2003, At. 73. [ONLINE: http://jipam.vu. edu.au/aticle.php?sid=314]. [7] J.E. PEČARIĆ, F. PROSCHAN AND Y.L. TONG, Convex, Patial Odeings, and Statistical Applications, Academic Pess, Inc. 1992. Page 26 of 27
[8] A. WITKOWSKI, A new poof of the monotonicity popety of powe means, J. Inequal. in Pue and Appl. Math., 53 2004, At. 73. [ONLINE: http://jipam.vu.edu.au/aticle.php?sid=425]. Page 27 of 27