Exponential convexity and Jensen's inequality for divided ... - Ele-Math

J ournal of Mathematical I nequalities Volume 5, Number 2 (2011), 157–168

EXPONENTIAL CONVEXITY AND JENSEN’S INEQUALITY FOR DIVIDED DIFFERENCES

Z. PAVI C´ , J. P E Cˇ ARI C´ AND A. V UKELI C´ Abstract. In this paper we obtain means which involve divided differences for n -convex functions. We examine their monotonicity property using exponentially convex functions.

1. Introduction Let f be a real-valued function defined on the segment [a, b]. The divided difference of order n of the function f at distinct points x0 , ..., xn ∈ [a, b], is defined recursively (see [3], [10] and [12]) by f [xi ] = f (xi ) for i = 0, . . . , n and

f [x1 , . . . , xn ] − f [x0 , . . . , xn−1 ] . xn − x0 The value f [x0 , . . . , xn ] is independent of the order of the points x0 , . . . , xn . The definition may be extended to include the case that some (or all) of the points coincide. Assuming that f ( j−1) (x) exists, we define f [x0 , . . . , xn ] =

f ( j−1) (x) f [x, . . . , x] = . ( j − 1)! j times

For divided difference, in the case of distinct points, the following holds: n

n f (xi ) , where ω (x) = ∏ (x − x j ), j=0 i=0 ω (xi )

f [x0 , . . . , xn ] = ∑ so we have that

n

f (xi ) . n i=0 ∏ j=1, j=i (xi − x j )

f [x0 , . . . , xn ] = ∑

A function f : [a, b] → R is said to be n -convex if the n -th order divided difference of f satisfies f [x0 , . . . , xn ] 0 for all a x0 < . . . < xn b. The following theorem is proved (see [5]): Mathematics subject classification (2010): 26D15, 26D20, 26D99. Keywords and phrases: Jensen inequality, divided differences, n -convex function, Schur convex function, log-convexity, exponentially convex function, majorization, continuous extensions, Schur means. c , Zagreb Paper JMI-05-14

157

158

ˇ ´ AND A. V UKELI C´ Z. PAVI C´ , J. P E CARI C

Then

T HEOREM 1. Let f be an (n + 2)-convex function on (a, b) and x ∈ (a, b)n+1 . G(x) = f [x0 , . . . , xn ]

is a convex function of the vector x = (x0 , . . . , xn ). Consequently, f

m

m

i=0

i=0

∑ aixi0 , . . . , ∑ ai xin

m

∑ ai f [xi0 , . . . , xin ]

(1.1)

i=0

holds for all ai 0 such that ∑m i=0 ai = 1 . Schur polynomial in n + 1 variables x0 , . . . , xn of degree d = d0 + . . . + dn ( d0 . . . dn ) is defined as d +j n det xi n− j i, j=0 n S(d0 ,...,dn ) (x0 , . . . , xn ) = . j det xi i, j=0

The numerator consists of alternating polynomials (they change the sign under any transposition of the variables) and so they are all divisible by the denominator which is Vandermonde determinant. Schur polynomial is also symmetric because the numerator and denominator are both alternating. For n 1 using Schur polinomial and Vandermonde determinant (extended with logarithmic function) ⎤ ⎡ 1 x0 x0 2 . . . x0 n−1 x0 p lnq x0 ⎢ 1 x1 x1 2 . . . x1 n−1 x1 p lnq x1 ⎥ ⎥ ⎢ 2 n−1 x p lnq x ⎥ ⎢ 2 2⎥ V (x; p, q) = det ⎢ 1 x2 x2 . . . x2 ⎥ ⎢ .. .. .. . . . .. ⎦ ⎣. . . . .. . q 2 n−1 p 1 xn xn . . . xn xn ln xn we obtain: P ROPOSITION 1. For monomial function h(x) = xn+k , where k 1 is an integer, holds V (x; n + k, 0) . h[x0 , . . . , xn ] = S(k,0, . . . , 0) (x0 , . . . , xn ) = V (x; n, 0) n times

For potential function f (x)

= xp

= xn+p−n , where p is a real number, holds

f [x0 , . . . , xn ] =

V (x; p, 0) . V (x; n, 0)

Further, for a partition π the Schur polynomial can be expressed by a sum Sπ (x0 , . . . , xn ) = ∑ xT = ∑ x00 . . . xtnn . t

T

T

159

E XPONENTIAL CONVEXITY AND J ENSEN ’ S INEQUALITY

The summation is over all semistandard Young tableaux T of shape π where the exponents t0 , . . .tn give the weight of T in which each t j counts the occurences of the number j in T (see [2] and [7]). So, we have the next proposition: P ROPOSITION 2. For monomial function h(x) = xn+k , where k 1 is an integer, holds ik−1

i1

n

h[x0 , . . . , xn ] = S(k,0, . . . , 0) (x0 , . . . , xn ) = ∑ ∑ . . . ∑ xi1 xi2 · · · xik . i1 =0 i2 =0 ik =0 n times

The goal of this paper is to define means using the inequality (1.1). We will examine their monotonicity property using exponential convex functions. Also, we will consider some special cases. 2. Jensen means for divided differences T HEOREM 2. Let f ∈ Cn+2 ([a, b]) and xi ∈ (a, b)n+1 for i = 0, . . . ,m. If ai j

n m m i i i m l 0 such that ∑m i=0 ai = 1 and ∑ j=0 ∑k=0 ∑i=0 ai x j xk − ∑i=0 ai x j ∑l=0 al xk = 0 , then there exists ξ ∈ (a, b) such that m

m

m

i=0

i=0

∑ ai f [xi0 , . . . , xin ] − f ∑ ai xi0 , . . . , ∑ ai xin

i=0

f (n+2) (ξ ) = (n + 2)!

n

j

m

∑∑ ∑

j=0 k=0

i=0

ai xij xik −

m

∑

i=0

ai xij

m

∑

l=0

al xlk

.

(2.1)

Proof. Let us denote α = min f (n+2) and β = max f (n+2) . We first consider the (n+2) β xn+2 following function φ1 (x) = (n+2)! − f (x). Then φ1 (x) = β − f (n+2) (x) 0, x ∈ [a, b], so φ1 is an (n + 2)-convex function. Applying Theorem 1 on an (n + 2)-convex xn+2 we have function φ1 with φ (x) = (n+2)! m

m

β · ∑ ai φ [xi0 , . . . , xin ] − ∑ ai f [xi0 , . . . , xin ] i=0

i=0

m

∑

β ·φ

i=0

ai xi0 , . . . ,

m

∑

i=0

ai xin

i.e. m

∑

i=0

ai f [xi0 , . . . , xin ] −

β

m

∑

i=0

f

−f

m

∑

i=0

m

∑

i=0

ai xi0 , . . . ,

ai φ [xi0 , . . . , xin ] − φ

ai xi0 , . . . ,

m

∑

i=0

m

∑

i=0

m

∑

i=0

ai xin

ai xin

ai xi0 , . . . ,

m

∑

i=0

ai xin

.

,

160


Similarly, a function φ2 (x) = f (x) − α · φ (x) is an (n + 2)-convex function. Inequality from the Theorem 1 with (n + 2)-convex function φ2 becomes m

m

∑ ai f [xi0 , . . . , xin ] − α · ∑ ai φ [xi0 , . . . , xin ]

i=0

i=0

m

m

i=0

i=0

∑ ai xi0, . . . , ∑ ai xin

f

m

m

i=0

i=0

∑ ai xi0, . . . , ∑ ai xin

−α ·φ

,

i.e.

m

m

m

∑ ai f [xi0 , . . . , xin ] − f ∑ ai xi0 , . . . , ∑ aixin

i=0

α

i=0

m

∑

i=0

ai φ [xi0 , . . . , xin ] − φ

i=0

m

∑

i=0

ai xi0 , . . . ,

m

∑

i=0

ai xin

.

From Proposition 2 with k = 2 and the fact that the function φ (x) is (n + 2)-convex we have m

∑ ai φ [xi0 , . . . , xin ] − φ

i=0

1 = (n + 2)!

n

j

m

m

∑ aixi0 , . . . , ∑ ai xin

i=0

i=0

m

m

m

i=0

i=0

l=0

∑ ∑ ∑ aixij xik − ∑ ai xij ∑ al xlk

j=0 k=0

> 0.

We can now conclude that there exists ξ ∈ (a, b) that we are looking for in (2.1).

R EMARK 1. Let us note that the left-hand side in equation (2.1) is greater than or equal to zero if f (n+2) 0 which is the statement of Theorem 1. C OROLLARY 1. Let f , g ∈ Cn+2 ([a, b]) and xi ∈ (a, b)n+1 . If ai 0 and ∑m i=0 ai = 1 , then there exists ξ ∈ (a, b) such that m m i i i i f (n+2) (ξ ) ∑m i=0 ai f [x0 , . . . , xn ] − f ∑i=0 ai x0 , . . . , ∑i=0 ai xn = (2.2) m m m i i g(n+2)(ξ ) ∑i=0 ai g[x0 , . . . , xin ] − g ∑i=0 ai x0 , . . . , ∑i=0 ai xin provided that both denominators not equal zero. Proof. We use the following technique: Let us define the linear functional standard m m i i i i L(h) = ∑m i=0 ai h[x0 , . . . , xn ] − h ∑i=0 ai x0 , . . . , ∑i=0 ai xn . Next, we define ψ (t) = f (t)L(g) − g(t)L( f ). According to Theorem 2, applied on ψ , there exists ξ ∈ (a, b) so that m m m ψ (n+2) (ξ ) n j i i i l L(ψ ) = ∑ ∑ ∑ a i x j xk − ∑ a i x j ∑ a l xk . (n + 2)! j=0 i=0 k=0 i=0 l=0


From L(ψ ) = 0 , it follows f (n+2) (ξ )L(g)−g(n+2)(ξ )L( f ) = 0 and (2.2) is proved.

161

The above corollary enables us to define various types of means, because if the function f (n+2) /g(n+2) has inverse, from (2.2) we can get

ξ=

f (n+2) g(n+2)

−1

m m i i i i ∑m i=0 ai f [x0 , . . . , xn ] − f ∑i=0 ai x0 , . . . , ∑i=0 ai xn m . m i i i i ∑m i=0 ai g[x0 , . . . , xn ] − g ∑i=0 ai x0 , . . . , ∑i=0 ai xn

Specially, if we take substitutions f (t) = t p , g(t) = t q with t > 0 in the above identity and use the method of continuous extensions, we get Jensen means for divided differences. Let us use the following notation: ⎡

x00

⎢ ⎢ 1 ⎢x ⎢ 0 X=⎢ . ⎢ . ⎢ . ⎣ xm 0

x01

x02

x11 .. .

x12 . . . x1n−1 .. . . .. . . .

...

x0n−1

m m xm 1 x2 . . . xn−1

x0n

⎤

⎥ ⎥ x1n ⎥ ⎥ , a = (a0 , . . . , am ), .. ⎥ ⎥ . ⎥ ⎦ xm n

V xpq = V (x; p, q) and N = {0, 1, . . . , n + 1} The Jensen means for divided differences is defined as

EJen (X, a; p, q) =

⎧ 1 j p−q ⎪ m a V x p0 − V aXp0 ⎪ ∑ j j=0 j V aXn0 ⎪ q−i n+1 V x n0 ⎪ for p = q; p ∈ / N, q ∈ /N ∏ ⎪ j i=0 p−i V x q0 V aXq0 ⎪ ⎪ ∑mj=0 a j j − V aXn0 ⎪ V x n0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ j k−q ⎪ aXk1 ⎪ ∑mj=0 a j V x j k1 − VV aXn0 ⎪ (q−k)(q−i) n+1 V x n0 ⎪ for p = q; p = k ∈ N, q ∈ /N ⎪ ∏ i=0 k−i ⎪ V x j q0 V aXq0 ⎪ ∑mj=0 a j j − V aXn0 i=k ⎪ V x n0 ⎪ ⎪ ⎪ ⎪ ⎪ 1 ⎪ j p−l ⎪ m a V x p0 − V aXp0 ⎪ n+1 ∑ j ⎪ j V aXn0 j=0 l−i V x n0 ⎪ for p = q; p ∈ / N, q = l ∈ N ⎪ ∏ j i=0 ⎨ (p−l)(p−i) m V x l1 V aXl1 ∑ j=0 a j

i=l

V x j n0

− V aXn0

⎪ ⎪ 1 ⎪ k−l ⎪ V x j k1 V aXk1 m ⎪ ⎪ n+1 l−i ∑ j=0 a j V x j n0 − V aXn0 ⎪ for p = q; p = k ∈ N, q = l ∈ N ⎪ ∏ j i=0 i−k m ⎪ aXl1 ⎪ i= ∑ j=0 a j V x j l1 − VVaXn0 ⎪ k,l V x n0 ⎪ ⎪ ⎪ ⎪ ⎪ j ⎪ m a V x p1 − V aXp1 ⎪ ∑ j j=0 j ⎪ V aXn0 1 V x n0 ⎪ for p = q; p ∈ / N, q ∈ /N exp ∑n+1 ⎪ i=0 i−p + m V x j p0 V aXp0 ⎪ ⎪ ∑ j=0 a j j − V aXn0 ⎪ V x n0 ⎪ ⎪ ⎪ ⎪ ⎪ j aXk2 ⎪ ⎪ ∑mj=0 a j V x j k2 − VV aXn0 ⎪ n+1 1 1 V x n0 ⎪ for p = q; p = q = k ∈ N ⎩exp ∑i=0 i−k + 2 m a V x j k1 − V aXk1 i=k

∑ j=0 j j V x n0

V aXn0

(2.3)

162


A mapping EJen : R × R → R is a continuous function in two variables p and q . The skeleton of all expressions for this means is a fraction m V aXp0 V xi p0 m i i i i ∑m ∑m i=0 ai V xi n0 − V aXn0 i=0 ai f [x0 , . . . , xn ] − f ∑i=0 ai x0 , . . . , ∑i=0 ai xn = i m m i i i i V aXq0 V x q0 ∑m i=0 ai g[x0 , . . . , xn ] − g ∑i=0 ai x0 , . . . , ∑i=0 ai xn ∑m i=0 ai V xi n0 − V aXn0 which follows from Proposition 1. The most parts of expressions proceed from V xi p0

V aXp0

m ∑m V xi k1 V aXk1 i=0 ai V xi n0 − V aXn0 lim = ∑ ai i − p→k p−k V aXn0 i=0 V x n0

for k ∈ N.

(2.4)

The quotient under limit becomes an indeterminate form when p → k ∈ N because V xi k0 V aXk0 V xk0 = 0 for k = 0, 1, ..., n − 1 and ∑m i=0 ai V xi n0 − V aXn0 = 1 − 1 = 0 for k = n . When p → k = n + 1: m

V xi k0

V aXk0

m

n

i=0

j=0

n

m

∑ ai V xi n0 − V aXn0 = ∑ ai ∑ xij − ∑ ∑ ai xij = 0.

i=0

j=0 i=0

So, we can apply L’Hospital’s rule and also the formula ddp V xpq = V x p q+1 to get result (2.4). If n = 0 the expressions for means EJen (X, a; p, q) proceed from the continuous extensions of function 1 m i p i p p−q q ∑m i=0 ai (x0 ) − (∑i=0 ai x0 ) . (p, q) → m i q i q p ∑m i=0 ai (x0 ) − (∑i=0 ai x0 ) 3. Monotonicity of Jensen means for divided differences Now we can introduce exponentially convex functions that will play an important role in this section. First, we give here a definition of exponentially convex function as it was done originally by Bernstein in [4] (see also [1],[8], [9]). D EFINITION 1. A function ϕ : (a, b) → R is exponentially convex if it is continn

uous and ∑ ξi ξ j ϕ (xi + x j ) 0 for every n ∈ N and all choices ξ1 , . . . , ξn ∈ R and i, j=1

all choices x1 , . . . , xn ∈ (a, b) so that xi + x j ∈ (a, b). In the rest of the paper we will rely on the following proposition and its corollaries. lent:

P ROPOSITION 3. Let ϕ : (a, b) → R. Then the following statements are equiva-

(i) ϕ is exponentially convex. (ii) ϕ is continuous and

n

xi + x j ∑ ξi ξ j ϕ 2 i, j=1

0

(3.1)

for every n ∈ N and all choices ξ1 , . . . , ξn ∈ R and all choices x1 , . . . , xn ∈ (a, b).

163


Using basic calculus, we have these two corollaries. C OROLLARY 2. If ϕ is exponentially convex, then xi + x j n 0 det ϕ 2 i, j=1 for every n ∈ N and all choices x1 , . . . , xn ∈ (a, b). C OROLLARY 3. Exponentially convex function ϕ : (a, b) → (0, ∞) is log-convex: x+y ϕ ϕ (x)ϕ (y) for all x, y ∈ (a, b). 2 Using exponential convexity we will prove monotonicity of Jensen means E(X, a; p, q) in both parameters p and q . i , . . . , xi for i = 0, . . . , m be mutually different positive real T HEOREM 3. Let xi0 , x n 1 j

m i i i m l numbers so that ∑nj=0 ∑k=0 ∑m i=0 ai x j xk − ∑i=0 ai x j ∑l=0 al xk = 0 for all ai 0 and m ∑i=0 ai = 1 . Also let m

ϕ (λ ) = ∑ ai fλ [xi0 , . . . , xin ] − fλ i=0

where fλ is defined with ⎧ tλ ⎪ ⎨ λ (λ −1)···( λ −n−1) fλ (t) = ⎪ t k lnt ⎩ n+1−k (−1)

k!(n+1−k)!

m

m

i=0

i=0

∑ aixi0 , . . . , ∑ ai xin

,

if λ ∈ / {0, 1, . . . , n + 1},

(3.2)

i f λ = k ∈ {0, 1, . . ., n + 1}.

(i) The function ϕ is exponentially convex. n t +t (ii) For every n ∈ N and all choices t1 , . . . ,tn ∈ R the matrix ϕ i 2 j positive semi-definite matrix. Particularly n t +t det ϕ i 2 j

i, j=1

i, j=1

0.

is a

(3.3)

Proof. (i) First, the expression ⎧ V (xi ;λ ,0) V (aX;λ ,0) 1 m ⎪ if λ ∈ a − / {0, 1, . . . , n + 1}, ⎨ λ (λ −1)···( ∑ i i i=0 λ −n−1) V (aX;n,0) V (x ;n,0) ϕ (λ ) = i ⎪ (aX;k,1) 1 m ⎩ i f λ = k ∈ {0, 1, . . . , n + 1}, a V (x ;k,1) − VV (aX;n,0) (−1)n+1−k k!(n+1−k)! ∑i=0 i V (xi ;n,0)

164


shows that ϕ is a continuous function. Further, let us consider the function f (t) = n

(n+2)

∑ ξi ξ j f ti +t j (t), where ξi ∈ R and ti ∈ R for i = 1, . . . , n . Since fλ

i, j=1

it follows

2

f

(n+2)

(t) =

n

∑

ξi ξ j t

ti +t j 2

−n−2

=

i, j=1

n

∑ ξi t

ti −n−2 2

(t) = t λ −n−2 ,

2 0.

i=1

m m i i i i This is equivalent to inequality ∑m i=0 ai f [x0 , . . . , xn ] − f ∑i=0 ai x0 , . . . , ∑i=0 ai xn 0 , i.e. n t +t (3.4) ∑ ξi ξ j ϕ i 2 j 0. i, j=1

We now conclude that ϕ is an exponentially convex function. (ii) This follows from nonnegativity of quadratic form (3.4).

L EMMA 1. If ϕ : (a, b) → (0, ∞) is log-convex and s,t, u, v ∈ (a, b) so that s = t, u = v, s u,t u (or s v,t v); then

ϕ (t) ϕ (s)

1 t−s

ϕ (v) ϕ (u)

1 v−u

.

C OROLLARY 4. Let p u and q v. Then EJen (X, a; p, q) EJen (X, a; u, v)

(3.5)

for all xi0 , xi1 , . . . , xin mutually different positive real numbers so that

j m i i i m l 0 for all a 0 and m a = 1 . ∑nj=0 ∑k=0 ∑m ∑i=0 i i i=0 ai x j xk − ∑i=0 ai x j ∑l=0 al xk =

Proof. Let us first observe that for p = q holds EJen (X, a; p, q) =

ϕ (p) ϕ (q)

1 p−q

.

(3.6)

According to Theorem 3 ϕ is a positive exponentially convex function and therefore, according to Corollary 3, a log-convex function. So, we can apply Lemma 1 on ϕ and get 1 1 ϕ (t) t−s ϕ (v) v−u (3.7) ϕ (s) ϕ (u) for s,t, u, v ∈ R so that s = t, u = v, s u,t u (or s v,t v). Using continuous extensions of (3.6) and (3.7) we finally conclude that, p u and q v implies EJen (X, a; p, q) EJen (X, a; u, v).


165

4. Special cases If in (2.3) we put xi0 = xi , xij = xi + h j and ai = Ppmi , where i = 0, . . . , m, Pm = j = 1, . . . , n and xi + h j ∈ (a, b), we get means related to inequality (see [16]):

∑m i=0 pi ,

1 m ∑ pi f [xi , xi + h1, . . . , xi + hn] f [x, x + h1, . . . , x + hn], Pm i=0 i where x = P1m ∑m i=0 pi x . If we put n = 1, h = xi1 − xi0 and xi = xi0 , yi = xi1 for i = 0, . . . , m in above case we get means related to inequality (see [13]): 1 m 1 m 1 m 1 m ∑ pi f (xi ) − f Pm ∑ pi xi Pm ∑ pi f (yi ) − f Pm ∑ pi yi . Pm i=0 i=0 i=0 i=0

If in (2.3) we put n = 1, xi0 = 2a − xi , xi1 = xi , where i = 0, . . . , m and xi ∈ [0, 2a], we get means related to inequality (see [15] and [12]): m

∑ ai

i=0

m f (∑m f (xi ) − f (2a − xi) i=0 ai xi ) − f (2a − ∑i=0 ai xi ) . m xi − a ∑i=0 ai xi − a

4.1. Shur means Let x = (x0 , . . . , xn ) and y = (y0 , . . . , yn ) denote two real (n + 1)-tuples. We say that x majorizes y and put x y if k

k

i=0

i=0

∑ x[i] ∑ y[i]

and

for k = 0, 1, . . . , n − 1

n

n

i=0

i=0

∑ xi = ∑ yi ,

where x[0] x[1] · · · x[n] and y[0] y[1] · · · y[n] are the nonincreasing ordered components of x and y . An important tool in the study of majorization is a theorem (see [6]) which says that for x, y ∈ Rn , x y if and only if y = xP for some double stohastic matrix P. A square matrix P is said to be stochastic if its elements are all nonnegative and all row sums are one. If in addition to being stochastic, all column sums are one, the matrix is said to be double stochastic. If we put m = n , ⎤ ⎡ a0 an an−1 . . . a2 a1 ⎢ a1 a0 an . . . a3 a2 ⎥ ⎥ ⎢ ⎢ .. .. . . .. .. ⎥ , P = ⎢ ... . . . . . ⎥ ⎥ ⎢ ⎣ an−1 an−2 an−3 . . . a0 an ⎦ an an−1 an−2 . . . a1 a0

166


and x0 x

1

= (x0 , x1 , . . . , xn−1 , xn ), = (x1 , x2 , . . . , xn , x0 ),

..., xn−1 = (xn−1 , xn , . . . , xn−3 , xn−2 ), xn

= (xn , x0 , . . . , xn−2 , xn−1 )

we have y = (y0 , y1 , . . . , yn )

⎡

a0 a1 .. .

an an−1 ⎢ a0 an ⎢ ⎢ .. .. = (x0 , x1 , . . . , xn ) · ⎢ . . ⎢ ⎣ an−1 an−2 an−3 an an−1 an−2 =

m

m

i=0

i=0

∑ aixi0 , . . . , ∑ ai xin

⎤ a1 a2 ⎥ ⎥ .. ⎥ . ⎥ ⎥ . . . a0 an ⎦ . . . a1 a0 ... ... .. .

a2 a3 .. .

= aX.

Now, using Theorem 1 we get inequality (see [14]): f [x0 , . . . , xn ] f [y0 , . . . , yn ], i.e. function G defined in Theorem 1 is Schur-convex. So, (2.3) in this case are Schur means (see [11]). The following result (see [5]) is a consequence of the previous result:

T HEOREM 4. Let f (n) be convex in (a, b), a x0 · · · xn b , and define 1 x = n+1 ∑ni=0 xi . Then f (n) (x) = f [x, . . . , x] f [x0 , . . . , xn ]. n!

(4.1)

n+1 times

If x0 = xn , then equality in (4.1) holds iff f ∈ Πn+1 where Πn+1 denotes the class of polynomials of degree at most n + 1 . For the Schur means related with above inequality we have the following:


167

ESch (x; p, q) = ⎧ 1 p−q n−1 p ⎪ V xp0 ∏i=0 (p−i)x ⎪ − ⎪ n+1 q−i V xn0 n!· x n ⎪ for p = q; p ∈ / N, q ∈ /N ⎪ ∏ ⎪ (q−i)x q ⎪ i=0 p−i V xq0 − ∏n−1 i=0 ⎪ n ⎪ V xn0 n!· x ⎪ 1 ⎞ k−q ⎪⎛ n−1 n−1 (k−i)x k +∏n−1 (k−i)x k ln x ⎪ ⎪ ∑ j=0 ∏ i=0 ⎪ i=0 ⎪ ⎪ i= j xk1 ⎟ ⎜ n+1 (q−k)(q−i) VV xn0 ⎪ − ⎪ n!· x n ⎟ ⎜∏ ⎪ for p = q; p = k ∈ N, q ∈ /N ⎪ n−1 (q−i)x q i=0 k−i ⎠ ⎝ ⎪ ∏ V xq0 ⎪ i=0 i=k − ⎪ V xn0 ⎪ n!· x n ⎪ ⎪ ⎪ ⎪ ⎛ ⎞ 1 ⎪ ⎪ p−l ⎪ ⎪ n−1 p ⎪ V xp0 ∏i=0 (p−i)x ⎪⎜ n+1 ⎟ − ⎪ l−i V xn0 n!· x n ⎪ ⎜ ⎟ for p = q; p ∈ / N, q = l ∈ N ⎪ n−1 n−1 (l−i)x l +∏n−1 (l−i)x l ln x ⎠ ⎪⎝∏i=0 (p−l)(p−i) ∑ j=0 ∏ ⎪ i=0 i=0 ⎪ i = l ⎪ i= j ⎪ V xl1 ⎪ ⎪ V xn0 − n!· x n ⎪ 1 ⎞ k−l ⎪ n−1 n−1 n−1 ⎨⎛ k k ∑ j=0 ∏

i=0

(k−i)x +∏

i=0

(k−i)x ln x

i= j xk1 − ⎜ n+1 l−i VV xn0 ⎟ ⎪ n!· x n ⎜∏ ⎟ for p = q; p = k ∈ N, q = l ∈ N ⎪ ⎪ n−1 n−1 (l−i)x l +∏n−1 (l−i)x l ln x ⎠ i=0 i−k ⎝ ⎪ ∑ ∏ ⎪ j=0 i=0 i=0 i = k,l ⎪ ⎪ i= j V xl1 ⎪ ⎪ V xn0 − n!· x n ⎪ ⎛ ⎞ ⎪ n−1 n−1 (p−i)x p +∏n−1 (p−i)x p ln x ⎪ ∑ j=0 ∏ ⎪ i=0 i=0 ⎪ ⎪ i= j V xp1 ⎜ n+1 1 ⎟ ⎪ − ⎪ n!· x n ⎟ for p = q; p ∈ ⎪ exp ⎜ / N, q ∈ /N ⎪ ∑i=0 i−p + V xn0 n−1 (p−i)x p ⎪ ⎝ ⎠ V xp0 ∏i=0 ⎪ ⎪ − n ⎪ V xn0 n!· x ⎪ ⎪ ⎪ ⎛ ⎪ n−1 n−1 n−1 n−1 n−1 n−1 k k k 2 ⎞ ⎪ ∑ ∑ ⎪ j1 =0 j2 =0 ∏ i=0 (k−i)x +2 ∑ j=0 ∏i=0 (k−i)x ln x +∏i=0 (k−i)x ln x ⎪ ⎪ i= j ⎪ j2 = j1 i= j1 ⎟ ⎜ ⎪ ⎪ i= j2 V xk2 − ⎟ ⎜ n+1 1 ⎪ n ⎪ 1 V xn0 n!· x ⎟ ⎜ ⎪ + exp ∑ ⎪ n−1 n−1 (k−i)x k +∏n−1 (k−i)x k ln x i=0 i−k 2 ⎟ ⎜ ⎪ ∑ ∏ j=0 i=0 i=0 ⎪ i=k ⎠ ⎝ ⎪ ⎪ i= j V xk1 − ⎪ ⎪ V xn0 n!· x n ⎪ ⎪ ⎩ for p = q; p = q = k ∈ N

Acknowledgement. The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grant 117 − 1170889 − 0888 . REFERENCES [1] N. I. A KHIEZER, The Classical Moment Problem and Some Related Questions in Analysis, Oliver and Boyd, Edinburgh, 1965. [2] G. A NDREWS , The Theory of Partitions, Encyclopedia Math. Appl. 12, Addison-Wesley, Reading, MA, 1976. [3] K. E. ATKINSON, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, 1989. [4] S. N. B ERNSTEIN, Sur les fonctions absolument monotones, Acta Math. 52 (1929), 1–66. [5] R. FARWIG , D. Z WICK, Some divided difference inequalities for n -convex functions, J. Math. Anal. Appl. 108 (1985), 430–437. ´ [6] G. H. H ARDY, J. E. L ITTLEWOOD , G. P OLYA , Some simple inequalities satisfied by convex functions, Messenger of Math 58 (1929), 145–152.

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[7] P. A. M ACMAHON, Combinatory Analysis, I, II, Chelsea, New York, 1960. ˇ ´ , On Some Inequalities for Monotone Functions, Boll. Unione. Mat. [8] D. S. M ITRINOVI C´ , J. E. P E CARI C Ital. (7) 5–13 (1991), 407–416. ˇ ´ AND A. M. F INK, Classical and new inequalities in analysis, Kluwer [9] D. S. M ITRINOVI C´ , J. P E CARI C Academic Publishers, The Netherlands, 1993. [10] A. DE M ORGAN, The Differential and Integral Calculus (Chapter XVIII, On Interpolation and Summation, page 550), Baldwin and Cradock, London, 1842. ˇ ´ , A. V UKELI C´ , Means for divided differences and exponential convexity, to [11] Z. PAVI C´ , J. P E CARI C appear in Mediterranean Journal of Mathematics. ˇ ´ , F. P ROSCHAN AND Y. L. T ONG, Convex functions, partial orderings, and statistical [12] J. E. P E CARI C applications, Mathematics in science and engineering, Vol. 187 Academic Press, 1992. ˇ ´ , An inequality for 3 -convex function, J. Math. Anal. Appl. 90 (1982), 213–218. [13] J. E. P E CARI C ˇ ´ , D. Z WICK, n -Convexity and Majorization, Rocky Mountain J. Math 19 (1989), 303– [14] J. E. P E CARI C 311. [15] P. M. VASI C´ , L J . R. S TANKOVI C´ , On some inequalities for convex and g -convex functions, Univ. Beograd. Publ. Elektrotehn. Fak. Se. Mat. Fiz. 412–460 (1973), 11–16. [16] D. Z WICK, A divided difference inequality for n -convex functions, J. Math. Anal. Appl. 104 (1984), 435–436.

(Received February 24, 2011)

Z. Pavić Mechanical Engineering Faculty University of Osijek Trg Ivane Brlić Maˇzuranić 2 35000 Slavonski Brod Croatia e-mail: [email protected] J. Peˇcarić Faculty of Textile Technology University of Zagreb Pierottijeva 6 10000 Zagreb, Croatia and Abdus Salam School of Mathematical Sciences GC University Lahore Gulberg 54660 Pakistan e-mail: [email protected] A. Vukelić Faculty of Food Technology and Biotechnology Mathematics department University of Zagreb Pierottijeva 6 10000 Zagreb Croatia e-mail: [email protected]

Journal of Mathematical Inequalities

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