DOI 10.1007/s11253-016-1185-6 Ukrainian Mathematical Journal, Vol. 67, No. 11, April, 2016 (Ukrainian Original Vol. 67, No. 11, November, 2015)

GENERALIZATIONS OF STEFFENSEN’S INEQUALITY BY LIDSTONE’S POLYNOMIALS J. Peˇcari´c1 , A. Peruˇsi´c2 , and K. Smoljak1

UDC 517.5

We obtain generalizations of Steffensen’s inequality by using Lidstone’s polynomials. Furthermore, the functionals associated with the obtained generalizations are used to generate n-exponentially and exponentially convex functions, as well as the new Stolarsky-type means.

1. Introduction Since its appearance in 1918, Steffensen’s inequality is still the subject of investigation and generalization by numerous mathematicians. The well-known Steffensen inequality reads [10]: Theorem 1.1. Suppose that f is decreasing and g is integrable on [a, b] with 0  g  1 and λ =

Then

Zb

f (t)dt 

b−λ

Zb a

f (t)g(t)dt 

Z

b

g(t)dt.

a

a+λ Z

f (t)dt.

a

The inequalities are reversed if f is increasing. In 1929, Lidstone [5] introduced a generalization of Taylor’s series, today known as the Lidstone series. It approximates a given function in the neighborhood of two points instead of one. These series were studied by Poritsky [8], Wittaker [13], Schoenberg [9], Boas [3], and other researchers. Definition 1.1. Let f 2 C 1 ([0, 1]), then the Lidstone series has the form ⌘ f (2k) (0)⇤k (1 − x) + f (2k) (1)⇤k (x) ,

1 ⇣ X k=0

where ⇤n is the Lidstone polynomial of degree 2n + 1 given by the formulas ⇤0 (t) = t, ⇤00n (t) = ⇤n−1 (t), ⇤n (0) = ⇤n (1) = 0, 1 2

University of Zagreb, Zagreb, Croatia. University of Rijeka, Rijeka, Croatia.

n ≥ 1.

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 67, No. 11, pp. 1525–1539, November, 2015. Original article submitted April 18, 2013. 0041-5995/16/6711–1721

c 2016 

Springer Science+Business Media New York

1721

ˇ ´ , A. P ERU Sˇ I C´ , J. P E CARI C

1722

AND

K. S MOLJAK

The other explicit representations for Lidstone polynomials can be found in [1] and [13]. Some of these representations are given by the formulas ⇤n (t) = (−1)

2

n

⇡ 2n+1

1 X (−1)k+1

k 2n+1

k=1

sin k⇡t,

n ≥ 1,

 � t2n−1 1 6t2n+1 ⇤n (t) = − 6 (2n + 1)! (2n − 1)! −

n−2 X k=0

2(22k+3 − 1) t2n−2k−3 B2k+4 , (2k + 4)! (2n − 2k − 3)!

22n+1 ⇤n (t) = B2n+1 (2n + 1)!

✓

1+t 2

where B2k+4 is the (2k + 4)th Bernoulli number and B2n+1

✓

In [12], Widder proved the following fundamental lemma:

◆

,

n = 1, 2, . . . ,

n = 1, 2, . . . ,

1+t 2

◆

is a Bernoulli polynomial.

Lemma 1.1. If f 2 C 2n ([0, 1]), then f (t) =

n−1 Xh

f

k=0

(2k)

(0)⇤k (1 − t) + f

(2k)

i

(1)⇤k (t) +

Z1

Gn (t, s)f (2n) (s)ds,

0

where

G1 (t, s) = G(t, s) =

8 <(t − 1)s :

for

s < t,

(s − 1)t for

t  s,

is the homogeneous Green function of the differential operator tions of G(t, s) :

Gn (t, s) =

Z1 0

d2 on [0, 1] with the following successive iterads2

G1 (t, p)Gn−1 (p, s)dp,

n ≥ 2.

The aim of the present paper is to generalize Steffensen’s inequality by using Lidstone’s polynomials. In Section 2, we obtain the difference of integrals on two intervals from which we get a general inequality. This general inequality is used in Section 3 to obtain new generalizations of Steffensen’s inequality for (2n)-convex functions. In Section 4, we estimate the difference of the left- and right-hand sides of the obtained generalizations. In Section 5, we consider three functionals associated with new generalizations and use them to generate n-exponentially and exponentially convex functions. In Section 6, we apply the results from Section 5 to some families of functions and obtain new Stolarsky-type means related to these functionals.

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2. Difference of Integrals on Two Intervals If [a, b] \ [c, d] 6= ?, we have four possible cases for two intervals [a, b] and [c, d]. The first case is [c, d] ⇢ [a, b], the second case is [a, b] \ [c, d] = [c, b], and the other two cases are obtained by changing a $ c and b $ d. Hence, in the following theorem we consider only the first two cases. [a,b] In the present paper, by Tw,n we denote

[a,b] Tw,n

=

n−1 X k=0

(b − a)

2k

Zb



w(x) f

a

(2k)

(a)⇤k

✓

b−x b−a

◆

+f

(2k)

(b)⇤k

✓

x−a b−a

◆�

dx.

Theorem 2.1. Let f : [a, b] [ [c, d] ! R be of the class C 2n on [a, b] [ [c, d ] for some n ≥ 1. Also let w : [a, b] ! [0, 1i and u : [c, d ] ! [0, 1i. If [a, b] \ [c, d] 6= ?, then Zb a

w(t)f (t)dt −

Zd c

u(t)f (t)dt −

[a,b] Tw,n

+

[c,d] Tu,n

=

max{b,d} Z

Kn (s)f (2n) (s)ds,

(2.1)

a

where, in the case [c, d] ✓ [a, b],

Kn (s) =

8 ◆ ✓ Zb > > > x−a s−a > 2n−1 > (b − a) dx, , w(x)Gn > > b−a b−a > > > a > > > ◆ ✓ Zb > > x−a s−a > 2n−1 > w(x)Gn dx , (b − a) > > < b−a b−a

s 2 [a, c],

a

◆ ✓ Zd > > x−c s−c > > 2n−1 > dx, s 2 hc, d], , u(x)Gn −(d − c) > > d−c d−c > > > c > > ◆ ✓ > Zb > > s − a x − a > 2n−1 > dx, s 2 hd, b], , w(x)Gn (b − a) > > b−a b−a :

(2.2)

a

and, in the case [a, b] \ [c, d] = [c, b],

Kn (s) =

8 ◆ ✓ Zb > > > x−a s−a > 2n−1 > (b − a) dx, , w(x)Gn > > b−a b−a > > > a > > > ◆ ✓ Zb > > x−a s−a > 2n−1 > dx , w(x)Gn (b − a) > > < b−a b−a

s 2 [a, c],

(2.3)

a

◆ ✓ Zd > > x−c s−c > > 2n−1 > dx, , u(x)Gn −(d − c) > > d−c d−c > > > c > > ◆ ✓ > Zd > > s − c x − c > 2n−1 > dx, , u(x)Gn −(d − c) > > d−c d−c : c

s 2 hc, b] , s 2 hb, d].

ˇ ´ , A. P ERU Sˇ I C´ , J. P E CARI C

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AND

K. S MOLJAK

Proof. From the Widder lemma, for f 2 C 2n ([a, b]), we get the following identity: f (x) =

n−1 X k=0

(b − a)

2k



f

(2k)

+ (b − a)

(a)⇤k

2n−1

Zb

✓

b−x b−a ✓

Gn

a

◆

+f

(2k)

(b)⇤k

✓

x−a b−a

◆�

◆ x − a s − a (2n) f , (s)ds. b−a b−a

(2.4)

Multiplying identity (2.4) by w(x), integrating from a to b, and using the Fubini theorem, we obtain Zb a

w(x)f (x)dx =

n−1 X k=0

(b − a)

2k

Zb



w(x) f

a

+ (b − a)2n−1

Zb a

(2k)

✓

(a)⇤k

b−x b−a

◆

+f

(2k)

✓

(b)⇤k

x−a b−a

1 0 b ◆ ✓ Z x−a s−a dxAds. , f (2n) (s)@ w(x)Gn b−a b−a

◆�

dx

(2.5)

a

Thus, subtracting identities (2.5) for the interval [a, b] and [c, d] we get (2.1). Theorem 2.2. Let f : [a, b] [ [c, d] ! R be (2n)-convex on [a, b] [ [c, d], let w : [a, b] ! [0, 1i, and let u : [c, d] ! [0, 1i. If [a, b] \ [c, d] 6= ? and Kn (s) ≥ 0,

(2.6)

then Zb a

w(t)f (t)dt −

[a,b] Tw,n

≥

Zd c

[c,d] u(t)f (t)dt − Tu,n ,

(2.7)

where, in the case [c, d] ✓ [a, b], Kn (s) is defined by (2.2) and, in the case [a, b] \ [c, d] = [c, b], Kn (s) is defined by (2.3). Proof. Since f is (2n)-convex, without loss of generality we can assume that f is (2n)-times differentiable and f (2n) ≥ 0 (see [7, p. 16 and 293]). We can now apply Theorem 2.1 to obtain (2.7). 3. Generalization of Steffensen’s Inequality by Lidstone’s Polynomials For a special choice of the weights and intervals in the previous section, we obtain the following generalization of Steffensen’s inequality: If

Theorem 3.1. Let f : [a, b][[a, a + λ] ! R be (2n)-convex on [a, b][[a, a+λ] and let w : [a, b] ! [0, 1i. Kn (s) ≥ 0,

(3.1)

G ENERALIZATIONS OF S TEFFENSEN ’ S I NEQUALITY BY L IDSTONE ’ S P OLYNOMIALS

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then Zb a

[a,b] w(t)f (t)dt − Tw,n ≥

a+λ Z a

[a,a+λ]

f (t)dt − T1,n

,

(3.2)

where, in the case a  a + λ  b, 8 ◆ ✓ Zb > > x−a s−a > 2n−1 >(b − a) dx , w(x)Gn > > b−a b−a > > > a > > a+λ > ◆ ✓ Z < x−a s−a 2n−1 Kn (s) = dx, , Gn −λ > λ λ > > a > > > b ◆ ✓ Z > > > x−a s−a > 2n−1 > dx, , w(x)Gn > :(b − a) b−a b−a a

s 2 [a, a + λ],

(3.3)

s 2 ha + λ, b] ,

and, in the case a < b  a + λ,

8 ◆ ✓ Zb > > x−a s−a > 2n−1 > (b − a) dx , w(x)Gn > > b−a b−a > > > a > > a+λ > ◆ ✓ Z < x−a s−a 2n−1 Kn (s) = Gn dx, s 2 [a, b], , −λ > λ λ > > a > > > a+λ ◆ ✓ Z > > > x−a s−a > 2n−1 > dx, s 2 hb, a + λ] . , Gn > :−λ λ λ

(3.4)

a

Proof. We take c = a, d = a + λ, and u(t) = 1 in Theorem 2.2. Theorem 3.2. Let f : [a, b] [ [b − λ, b] ! R be (2n)-convex on [a, b] [ [b − λ, b] and let w : [a, b] ! [0, 1i. If Kn (s) ≥ 0,

(3.5)

then Zb

b−λ

f (t)dt −

[b−λ,b] T1,n

≥

Zb a

[a,b] w(t)f (t)dt − Tw,n ,

(3.6)

ˇ ´ , A. P ERU Sˇ I C´ , J. P E CARI C

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AND

K. S MOLJAK

where, in the case a  b − λ  b,

8 ◆ ✓ Zb > > > x−a s−a > 2n−1 > dx, , −(b − a) w(x)Gn > > b−a b−a > > > a > > > ◆ ✓ Zb < x−b+λ s−b+λ 2n−1 dx , Gn Kn (s) = λ > λ λ > > b−λ > > > ◆ ✓ > Zb > > x−a s−a > 2n−1 > dx, , w(x)Gn −(b − a) > > b−a b−a : a

s 2 [a, b − λ], (3.7)

s 2 hb − λ, b] ,

and, in the case b − λ  a  b,

8 ◆ ✓ Zb > > x−b+λ s−b+λ > 2n−1 > > dx, s 2 [b − λ, a], , Gn λ > > λ λ > > > b−λ > > > ◆ ✓ Zb < x−b+λ s−b+λ 2n−1 Kn (s) = λ dx , Gn > λ λ > > > b−λ > > > ◆ ✓ Zb > > x−a s−a > 2n−1 > dx, s 2 ha, b] . , w(x)Gn −(b − a) > > : b−a b−a

(3.8)

a

Proof. First, we change a $ c, b $ d, and w $ u in Theorem 2.2 and then we take c = b − λ, d = b, and u(t) = 1. 4. Estimation of the Difference Theorem 4.1. Suppose that all assumptions of Theorem 2.1 are satisfied. Assume that (p, q) is a pair of conjugate exponents, i.e., 1  p, q  1, 1/p + 1/q = 1. Let � � � (2n) �p � : [a, b] [ [c, d] ! R �f be an R-integrable function for some n ≥ 1. Then � b � �Z � Zd � � [a,b] [c,d] � w(t)f (t)dt − u(t)f (t)dt − Tw,n + Tu,n � � � � � a

c

0

� � � � B  �f (2n) � @ p

The constant p = 1.

Z

max{b,d} a

|Kn (s)|q ds

!1/q

max{b,d} Z a

11/q

C |Kn (s)|q dsA

.

(4.1)

in inequality (4.1) is sharp for 1 < p  1 and best possible for

G ENERALIZATIONS OF S TEFFENSEN ’ S I NEQUALITY BY L IDSTONE ’ S P OLYNOMIALS

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Proof. By using inequality (2.1) and H¨older’s inequality, we obtain � b � �Z � Zd � � [a,b] [c,d] � w(t)f (t)dt − u(t)f (t)dt − Tw,n + Tu,n � � � � � a

c

� � 11/q 0 � max{b,d} � max{b,d} Z Z � � � � � C � � � B =� Kn (s)f (2n) (s)ds�  �f (2n) � @ |Kn (s)|q dsA . � � p � a � a

To prove the sharpness of the constant equality is realized in (4.1). For 1 < p < 1, we choose f such that

Z

max{b,d}

a

q

|Kn (s)| ds

!1/q

, we find a function f for which the

1

f (2n) (s) = sgn Kn (s) |Kn (s)| p−1 . For p = 1, we set f (2n) (s) = sgn Kn (s). For p = 1, we prove that � � 0 1 � � max{b,d} max{b,d} Z � � � � Z � (2n) � C B � � Kn (s)f (2n) (s)ds�  |Kn (s)| @ (s)� dsA max �f � � s2[a,max{b,d}] � � � a a

(4.2)

is the best possible inequality. Suppose that |Kn (s)| attains its maximum at s0 2 [a, max{b, d}]. First, we assume that Kn (s0 ) > 0. For sufficiently small ", we define f" (s) as follows: 8 0, > > > > > > < 1 (s − s0 )2n , f" (s) = " (2n)! > > > > 1 > > : (s − s0 )2n−1 , (2n)!

a  s  s0 , s0  s  s0 + ", s0 + "  s  max{b, d}.

Thus, for sufficiently small ", we get

� � � � � max{b,d} � � sZ0 +" sZ0 +" � Z � � � 1 �� 1 � � � (2n) Kn (s)f (s)ds� = � Kn (s) ds� = Kn (s) ds. � � � � " � " � a � s0 s0

Further, it follows from inequality (4.2) that 1 "

sZ0 +" s0

Kn (s)ds  Kn (s0 )

sZ0 +" s0

1 ds = Kn (s0 ). "

ˇ ´ , A. P ERU Sˇ I C´ , J. P E CARI C

1728

AND

K. S MOLJAK

Since 1 lim "!0 "

sZ0 +"

Kn (s)ds = Kn (s0 )

s0

the required assertion follows. For Kn (s0 ) < 0, we define 8 1 > > (s − s0 − ")2n−1 , > > (2n)! > > > < f" (s) = − 1 (s − s0 − ")2n , > > " (2n)! > > > > > :0,

a  s  s0 , s0  s  s0 + ", s0 + "  s  max{b, d},

and the remaining part of the proof is the same as above.

Theorem 4.2. Suppose that all assumptions of Theorem� 3.1 are �p satisfied. Assume that (p, q) is a pair of conjugate exponents, i.e., 1  p, q  1, 1/p + 1/q = 1. Let �f (2n) � : [a, b] [ [a, a + λ] ! R be an R-integrable function for some n ≥ 1. Also let Kn (s) be defined by (3.3) in the case a  a + λ  b and by (3.4) in the case a < b  a + λ. Then 11/q 0 � � b max{b,a+λ} a+λ � � �Z Z Z � � C � (2n) � B [a,a+λ] �� [a,b] � w(t)f (t)dt − f (t)dt − Tw,n + T1,n |Kn (s)|q dsA . � @ �  �f � p � � a

The constant for p = 1.

a

Z

a

max{b,a+λ}

q

|Kn (s)| ds

a

(4.3)

!1/q

in inequality (4.3) is sharp for 1 < p  1 and best possible

Proof. We take c = a, d = a + λ, and u(t) = 1 in Theorem 4.1. Theorem 4.3. Suppose that all assumptions of Theorem 3.2 are satisfied. Assume that (p, q) is a pair of conjugate exponents, i.e., 1  p, q  1, 1/p + 1/q = 1. Let � � � (2n) �p �f � : [a, b] [ [b − λ, b] ! R

be an R-integrable function for some n ≥ 1. Also let Kn (s) be defined by (3.7) in the case a  b − λ  b and by (3.8) in the case b − λ  a  b. Then 0 � � b � � �Z Zb � � � (2n) � B � [b−λ,b] [a,b] � � f (t)dt − w(t)f (t)dt − T + T � @ w,n 1,n �  �f � p � � a

b−λ

The constant p = 1.

Z

b

min{a,b−λ}

Zb

min{a,b−λ}

|Kn (s)|q ds

!1/q

11/q

C |Kn (s)|q dsA

.

(4.4)

in inequality (4.4) is sharp for 1 < p  1 and best possible for

G ENERALIZATIONS OF S TEFFENSEN ’ S I NEQUALITY BY L IDSTONE ’ S P OLYNOMIALS

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Proof. First, we change a $ c, b $ d, and w $ u in Theorem 2.1. Then we set c = b − λ,

d = b,

and

u(t) = 1.

The remaining part of the proof is similar to the proof of Theorem 4.1. 5. Mean-Value Theorems and Exponential Convexity Motivated by inequalities (2.7), (3.2), and (3.6) under the assumptions of Theorems 2.2, 3.1, and 3.2, respectively, we define the following linear functionals: L1 (f ) =

Zb

w(t)f (t)dt −

a

−

n−1 X

+

n−1 X

k=0

k=0

L2 (f ) =

Zb

(b − a)

(d − c)

−

n−1 X

+

n−1 X

k=0

λ

2k

k=0

Zb

2k

b−λ

−

+

k=0

n−1 X k=0

Zd



(2k)

w(x) f

u(x) f

c

λ

2k

(c)⇤k

✓

◆

b−x b−a

d−x d−c

◆

(b)⇤k

✓

x−a b−a

(d)⇤k

✓

x−c d−c

(b)⇤k

✓

x−a b−a

(a + λ)⇤k

✓

+f

+f

(2k)

(2k)

◆�

◆�

dx

dx,

(5.1)

f (t)dt

a

Zb

2k



w(x) f

a

a+λ Z 

Zb

(a)⇤k

✓

a+λ Z

f

(2k)

(a)⇤k

a

f (t)dt −

n−1 X

(2k)

a



Zb

2k

(b − a)

u(t)f (t)dt

c

w(t)f (t)dt −

a

L3 (f ) =

Zd

(2k)

✓

(a)⇤k

✓

◆

b−x b−a

a+λ−x λ

◆

+f

+f

(2k)

(2k)

x−a λ

◆�

◆�

dx

dx,

(5.2)

w(t)f (t)dt

a

Zb 

f

(2k)

(b − λ)⇤k

b−λ

(b − a)

2k

Zb a



w(x) f

(2k)

✓

b−x λ

(a)⇤k

✓

◆

+f

b−x b−a

(2k)

◆

(b)⇤k

+f

(2k)

✓

x−b+λ λ

(b)⇤k

✓

◆�

x−a b−a

dx

◆�

dx.

(5.3)

ˇ ´ , A. P ERU Sˇ I C´ , J. P E CARI C

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K. S MOLJAK

We also define I1 = [a, b] [ [c, d],

and

I2 = [a, b] [ [a, a + λ],

I3 = [a, b] [ [b − λ, b].

Remark 5.1. Under the assumptions of Theorems 2.2, 3.1, and 3.2 respectively, we get Li (f ) ≥ 0,

i = 1, 2, 3,

for all (2n)-convex functions f. First, we state and prove the mean-value theorems for the functionals thus defined. Theorem 5.1. Let f : Ii ! R, i = 1, 2, 3, be such that f 2 C 2n (Ii ). If inequalities in (2.6), i = 1, (3.1), i = 2, and (3.5), i = 3, hold, then there exist ⇠i 2 Ii such that Li (f ) = f (2n) (⇠i )Li ('),

(5.4)

i = 1, 2, 3,

where '(x) =

x2n . (2n)!

Proof. We denote m = min f (2n) and M = max f (2n) . For a given function f 2 C 2n (Ii ), we define functions F1 , F2 : Ii ! R with F1 (x) =

M x2n − f (x) (2n)!

and

F2 (x) = f (x) −

Thus, (2n)

F1

(x) = M − f (2n) (x) ≥ 0,

x 2 Ii

and, hence, we conclude that Li (F1 ) ≥ 0 and, therefore, Li (f )  M · Li ('). (2n)

Similarly, it follows from F2

(x) = f (2n) (x) − m ≥ 0 that m · Li (')  Li (f ).

If Li (') = 0, then (5.4) holds for all ⇠i 2 Ii . Otherwise, m

Li (f )  M. Li (')

Since f (2n) (x) is continuous on Ii , there exist ⇠i 2 Ii such that (5.4) holds.

mx2n . (2n)!

G ENERALIZATIONS OF S TEFFENSEN ’ S I NEQUALITY BY L IDSTONE ’ S P OLYNOMIALS

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Theorem 5.2. Let f, g : Ii ! R, i = 1, 2, 3, be such that f, g 2 C 2n (Ii ) and g (2n) (x) 6= 0 for every x 2 Ii . If inequalities in (2.6), i = 1, (3.1), i = 2, and (3.5), i = 3, hold, then there exist ⇠i 2 Ii such that Li (f ) f (2n) (⇠i ) = (2n) , Li (g) g (⇠i )

(5.5)

i = 1, 2, 3.

Proof. We define functions φi (x) = f (x)Li (g) − g(x)Li (f ),

i = 1, 2, 3.

Applying Theorem 5.1 for φi , we conclude that there exist ⇠i 2 Ii such that (2n)

Li (φi ) = φi

(⇠i )Li (').

Since Li (φi ) = 0, we get f (2n) (⇠i )Li (g) − g (2n) (⇠i )Li (f ) = 0 and, hence, (5.5) is proved. We now apply the functionals defined above to the construction of exponentially convex functions. We start this part of the section by formulating some definitions and properties used to get our results (see [6]). Definition 5.1. A function

: I ! R is n-exponentially convex in Jensen’s sense on I if n X

⇠i ⇠j

i,j=1

✓

xi + x j 2

◆

≥ 0,

holds for all choices ⇠1 , . . . , ⇠n 2 R and all choices x1 , . . . , xn 2 I. A function convex if it is n-exponentially convex in Jensen’s sense and continuous on I.

: I ! R is n-exponentially

Remark 5.2. It is clear from the definition that 1-exponentially convex functions in Jensen’s sense are, in fact, nonnegative functions. Moreover, the n-exponentially convex functions in Jensen’s sense are k-exponentially convex in Jensen’s sense for every k 2 N, k  n. Definition 5.2. A function : I ! R is exponentially convex in Jensen’s sense on I if it is n-exponentially convex in Jensen’s sense for all n 2 N. A function : I ! R is exponentially convex if it is exponentially convex in Jensen’s sense and continuous. Remark 5.3. It is known that

: I ! R is log-convex in Jensen’s sense if and only if ↵

2

(x) + 2↵β

✓

x+y 2

◆

+ β 2 (y) ≥ 0

holds for every ↵, β 2 R and x, y 2 I. Hence, a positive function is log-convex in Jensen’s sense if and only if it is 2-exponentially convex in Jensen’s sense. A positive function is log-convex if and only if it is 2-exponentially convex.

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Proposition 5.1. If f is a convex function on I and if x1  y1 , x2  y2 , x1 6= x2 , and y1 6= y2 , then the following inequality is true: f (x2 ) − f (x1 ) f (y2 ) − f (y1 )  . x2 − x1 y2 − y1 If the function f is concave, then this inequality is reversed. Lemma 5.1. A function Φ is log-convex on an interval I if and only if, for all a, b, c 2 I, a < b < c, the following inequality is true: [Φ(b)]c−a  [Φ(a)]c−b [Φ(c)]b−a . Definition 5.3. 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 2 [a, b] is defined recursively (see [2, 7]) by f [xi ] = f (xi ),

i = 0, . . . , n,

and f [x0 , . . . , xn ] =

f [x1 , . . . , xn ] − f [x0 , . . . , xn−1 ] . xn − x0

The value f [x0 , . . . , xn ] is independent of the order of the points x0 , . . . , xn . This definition can be extended to include the case in which some (or all) points coincide. Under the assumption that f (j−1) (x) exists, we define f (j−1) (x) . f [x, . . . , x] = | {z } (j − 1)!

(5.6)

j times

An elegant method of producing n-exponentially convex and exponentially convex functions is presented in [4]. We use this method to prove the n-exponential convexity for the functionals defined above. In what follows, the symbol log denotes the natural logarithm. Theorem 5.3. Let ⌦ = {fp : p 2 J}, where J is an interval in R, be a family of functions defined on an interval Ii , i = 1, 2, 3, in R such that the function p 7! fp [x0 , . . . , x2m ] is n-exponentially convex in Jensen’s sense on J for every (2m + 1) mutually different points x0 , . . . , x2m 2 Ii , i = 1, 2, 3. Let Li , i = 1, 2, 3, be linear functionals defined by (5.1)–(5.3). Then p 7! Li (fp ) is an n-exponentially convex function in Jensen’s sense on J. If the function p 7! Li (fp ) is continuous on J, then it is n-exponentially convex on J. Proof. For ⇠j 2 R, j = 1, . . . , n, and pj 2 J, j = 1, . . . , n, we define a function g(x) =

n X

⇠j ⇠k f pj +pk (x). 2

j,k=1

Under the assumption that the function p 7! fp [x0 , . . . , x2m ] is n-exponentially convex in Jensen’s sense, we get g[x0 , . . . , x2m ] =

n X

j,k=1

⇠j ⇠k f pj +pk [x0 , . . . , x2m ] ≥ 0. 2

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This, in turn, implies that g is a (2m)-convex function on J and, therefore, Li (g) ≥ 0, i = 1, 2, 3. Hence, n X

j,k=1

✓ ◆ ⇠j ⇠k Li f pj +pk ≥ 0. 2

We conclude that the function p 7! Li (fp ) is n-exponentially convex on J in Jensen’s sense. If the function p 7! Li (fp ) is also continuous on J, then p 7! Li (fp ) is n-exponentially convex by the definition. The following corollaries are immediate consequences of Theorem 5.3: Corollary 5.1. Let ⌦ = {fp : p 2 J}, where J is an interval in R, be a family of functions defined on an interval Ii , i = 1, 2, 3, in R such that the function p 7! fp [x0 , . . . , x2m ] is exponentially convex in Jensen’s sense on J for every (2m + 1) mutually different points x0 , . . . , x2m 2 Ii , i = 1, 2, 3. Let Li , i = 1, 2, 3, be linear functionals defined by (5.1)–(5.3). Then p 7! Li (fp ) is an exponentially convex function in Jensen’s sense on J. If the function p 7! Li (fp ) is continuous on J, then it is exponentially convex on J. Corollary 5.2. Let ⌦ = {fp : p 2 J}, where J is an interval in R, be a family of functions defined on an interval Ii , i = 1, 2, 3, in R such that the function p 7! fp [x0 , . . . , x2m ] is 2-exponentially convex in Jensen’s sense on J for every (2m + 1) mutually different points x0 , . . . , x2m 2 Ii , i = 1, 2, 3. Let Li , i = 1, 2, 3, be linear functionals defined by (5.1)–(5.3). Then the following statements are true: (i) If the function p 7! Li (fp ) is continuous on J, then it is a 2-exponentially convex function on J. If p 7! Li (fp ) is, in addition, strictly positive, then it is also log-convex on J. Furthermore, the following inequality is true: [Li (fs )]t−r  [Li (fr )]t−s [Li (ft )]s−r

(5.7)

for every choice of r, s, t 2 J such that r < s < t. (ii) If the function p 7! Li (fp ) is strictly positive and differentiable on J, then, for every p, q, u, v 2 J such that p  u and q  v, (5.8)

µp,q (Li , ⌦)  µu,v (Li , ⌦), where

for fp , fq 2 ⌦.

8 ✓ ◆ 1 > > p−q L (f ) i p > > , > > > Li (fq ) > > < 0 1 µp,q (Li , ⌦) = d > L (f ) > B dp i p C > > B C > exp > > @ Li (fp ) A , > > :

p 6= q, (5.9) p = q,

Proof. (i) This is an immediate consequence of Theorem 5.3, Remark 5.3, and Lemma 5.1.

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(ii) Since p 7! Li (fp ) is continuous and strictly positive, by (i), we conclude that p 7! Li (fp ) is log-convex on J, i.e., the function p 7! log Li (fp ) is convex on J. Applying Proposition 5.1, we get log Li (fp ) − log Li (fq ) log Li (fu ) − log Li (fv )  p−q u−v

(5.10)

for p  u, q  v, p 6= q, u 6= v. Hence, we conclude that µp,q (Li , ⌦)  µu,v (Li , ⌦). The cases p = q and u = v follow from (5.10) as the limit cases. Remark 5.4. Note that the results of Theorem 5.3 and Corollaries 5.1 and 5.2 remain valid when two points from the set x0 , . . . , x2m 2 Ii , i = 1, 2, 3, coincide, say x1 = x0 , for a family of differentiable functions fp such that the function p 7! fp [x0 , . . . , x2m ] is n-exponentially convex in Jensen’s sense (exponentially convex in Jensen’s sense, log-convex in Jensen’s sense). Moreover, they remain true when all 2m + 1 points coincide for a family of 2m differentiable functions with the same property. The proofs use (5.6) and suitable characterizations of convexity. 6. Applications to Stolarsky-Type Means In this section, we apply the general results obtained in the previous section to several families of functions satisfying the conditions of the established general results in order to get some other exponentially convex functions and Stolarsky means. Example 6.1. Consider a family of functions ⌦1 = {fp : R ! [0, 1) : p 2 R} defined by

fp (x) =

Here,

8 px e > > > p2n , <

> > x2n > : , (2n)!

p 6= 0, p = 0.

d2n fp (x) = epx > 0. dx2n This means that fp is (2n)-convex on R for every p 2 R and p 7!

d2n fp (x) dx2n

is exponentially convex by the definition. By using the same reasoning as in the proof of Theorem 5.3, we also conclude that p 7! fp [x0 , . . . , x2m ] is exponentially convex (and, hence, exponentially convex in Jensen’s sense).

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In view of Corollary 5.1, we conclude that p 7! Li (fp ), i = 1, 2, 3, are exponentially convex in Jensen’s sense. It is easy to see that this mapping is continuous (although the mapping p 7! fp is not continuous for p = 0). Thus, it is exponentially convex. For this family of functions, µp,q (Li , ⌦1 ), i = 1, 2, 3, it follows from (5.9) that 8 ◆ 1 ✓ > > p−q (f ) L > i p > , > > > Li (fq ) > > > < ◆ ✓ µp,q (Li , ⌦1 ) = exp Li (id · fp ) − 2n , > > Li (fp ) p > > > > ◆ ✓ > > 1 Li (id · f0 ) > > , :exp 2n + 1 Li (f0 )

p 6= q, p = q 6= 0, p = q = 0,

where id is the identity function. Moreover, by Corollary 5.2, it is a monotonic function of the parameters p and q. Theorem 5.2 applied to the functions fp , fq 2 ⌦1 and functionals Li , i = 1, 2, 3, implies that there exist ⇠i 2 Ii such that e(p−q)⇠i =

Li (fp ) . Li (fq )

Thus, Mp,q (Li , ⌦1 ) = log µp,q (Li , ⌦1 ),

i = 1, 2, 3,

satisfies the inequalities min{a, b − λ, c}  Mp,q (Li , ⌦1 )  max{a + λ, b, d},

i = 1, 2, 3.

Hence, Mp,q (Li , ⌦1 ) is the monotonic mean. Example 6.2. Consider a family of functions ⌦2 = {gp : (0, 1) ! R : p 2 R} defined by

gp (x) =

Here,

8 xp > > , > < p(p − 1) . . . (p − 2n + 1) > > > :

xj log x , (−1)2n−1−j j!(2n − 1 − j)!

p2 / {0, 1, . . . , 2n − 1}, p = j 2 {0, 1, . . . , 2n − 1}.

d2n gp (x) = xp−2n > 0, dx2n d2n gp (x) is exponentially convex by the definition. dx2n Arguing as in Example 6.1, we conclude that the mappings p 7! Li (gp ), i = 1, 2, 3, are exponentially convex. which means that gp is (2n)-convex for x > 0 and p 7!

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For this family of functions µp,j (Li , ⌦2 ), i = 1, 2, 3, from (5.9), we now get 8 ✓ ◆ 1 > > Li (gp ) p−q > > , > > > Li (gq ) > > > > ! > 2n−1 > X 1 > (g g ) L < i 0 p + , exp (−1)2n−1 (2n − 1)! µp,q (Li , ⌦2 ) = Li (gp ) i−p > i=0 > > 1 0 > > > 2n−1 > X 1 C > Li (g0 gp ) > B > + exp@(−1)2n−1 (2n − 1)! > A, > > 2Li (gp ) i−p > i=0 :

p 6= q, p=q2 / {0, 1, . . . , 2n − 1}, p = q 2 {0, 1, . . . , 2n − 1}.

i6=p

By using Theorem 5.2 once again, we conclude that

min{a, b − λ, c} 

✓

Li (gp ) Li (gq )

◆

1 p−q

 max{a + λ, b, d},

i = 1, 2, 3.

Hence, µp,q (Li , ⌦2 ), i = 1, 2, 3 is the mean. Example 6.3. Consider a family of functions ⌦3 = {φp : (0, 1) ! (0, 1) : p 2 (0, 1)} defined by

φp (x) =

Since

8 p−x > > > < (log p)2n , > > x2n > : , (2n)!

p 6= 1, p = 1.

d2n φp (x) = p−x dx2n is the Laplace transform of a nonnegative function (see [11]), it is exponentially convex. Obviously, φp are (2n)-convex functions for every p > 0. For this family of functions, the functions µp,q (Li , ⌦3 ), i = 1, 2, 3, from (5.9) are equal to 8 ◆ 1 ✓ > > > Li (φp ) p−q > , > > > Li (φq ) > > > < ◆ ✓ µp,q (Li , ⌦3 ) = exp − Li (id · φp ) − 2n , > > p Li (φp ) p log p > > > > ◆ ✓ > > 1 Li (id · φ1 ) > > , :exp − 2n + 1 Li (φ1 )

p 6= q, p = q 6= 1, p = q = 1,

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where id is the identity function. This function is monotone in the parameters p and q by (5.8). By virtue of Theorem 5.2, we conclude that Mp,q (Li , ⌦3 ) = −L(p, q) log µp,q (Li , ⌦3 ),

i = 1, 2, 3,

satisfies min{a, b − λ, c}  Mp,q (Li , ⌦3 )  max{a + λ, b, d}. Hence, Mp,q (Li , ⌦3 ) is the monotonic mean and L(p, q) is the logarithmic mean defined by

L(p, q) =

8 > <

p−q , log p − log q

> :p,

p 6= q, p = q.

Example 6.4. Consider a family of functions ⌦4 = {

p:

(0, 1) ! (0, 1) : p 2 (0, 1)}

defined by p

e−x p . p (x) = pn Since p d2n p (x) = e−x p 2n dx

is the Laplace transform of a nonnegative function (see [11]), it is exponentially convex. Clearly, p are (2n)-convex functions for every p > 0. For this family of functions, the functions µp,q (Li , ⌦4 ), i = 1, 2, 3, from (5.9) are equal to 8✓ ◆ 1 > Li ( p ) p−q > > > , p 6= q, < L( ) i q µp,q (Li , ⌦4 ) = ◆ ✓ > > Li (id · p ) n > > :exp − p , p = q, − 2 pLi ( p ) p

where id is the identity function. This function is monotone in the parameters p and q by (5.8). In view of Theorem 5.2, ⇣p p ⌘ Mp,q (Li , ⌦4 ) = − p + q log µp,q (Li , ⌦4 ), i = 1, 2, 3, satisfies the inequalities

min{a, b − λ, c}  Mp,q (Li , ⌦4 )  max{a + λ, b, d} and, hence, Mp,q (Li , ⌦4 ) is the monotonic mean.

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The research of the authors was supported by the Ministry of Science, Education, and Sports, under the Research Grants 117-1170889 (the first and third authors) and 114-0000000-3145 (the second author). REFERENCES 1. R. P. Agarwal and P. J. Y. Wong, Error Inequalities in Polynomial Interpolation and Their Applications, Kluwer, Dordrecht, etc. (1993). 2. K. E. Atkinson, An Introduction to Numerical Analysis, 2nd ed., Wiley, New York, etc. (1989). 3. R. P. Boas, “Representation of functions by Lidstone series,” Duke Math. J., 10, 239–245 (1943). 4. J. Jakˇseti´c and J. Peˇcari´c, “Exponential convexity method,” J. Convex Anal., 20, No. 1, 181–197 (2013). 5. G. J. Lidstone, “Notes on the extension of Aitken’s theorem (for polynomial interpolation) to the Everett types,” Proc. Edinburgh Math. Soc., 2, No. 2, 16–19 (1929). 6. J. Peˇcari´c and J. Peri´c, “Improvement of the Giaccardi and the Petrovi´c inequality and related Stolarsky-type means,” An. Univ. Craiova Ser. Mat. Inform., 39, No. 1, 65–75 (2012). 7. J. E. Peˇcari´c, F. Proschan, and Y. L. Tong, “Convex functions, partial orderings, and statistical applications,” Math. Sci. Eng., 187 (1992). 8. H. Poritsky, “On certain polynomial and other approximations to analytic functions,” Trans. Amer. Math. Soc., 34, No. 2, 274–331 (1932). 9. I. J. Schoenberg, “On certain two-point expansions of integral functions of exponential type,” Bull. Amer. Math. Soc., 42, No. 4, 284–288 (1936). 10. J. F. Steffensen, “On certain inequalities between mean values and their application to actuarial problems,” Skand. Aktuarietidskr., 82–97 (1918). 11. D. V. Widder, The Laplace Transform, Princeton Univ. Press, New Jersey (1941). 12. D. V. Widder, “Completely convex functions and Lidstone series,” Trans. Amer. Math. Soc., 51, 387–398 (1942). 13. J. M. Whittaker, “On Lidstone series and two-point expansions of analytic functions,” Proc. London Math. Soc., 36, 451–469 (1933-1934).