Acta Math Vietnam (2014) 39:347–358 DOI 10.1007/s40306-014-0067-y

STOLARSKY-TYPE MEANS RELATED TO GENERALIZATIONS OF STEFFENSEN’S AND GAUSS’ INEQUALITY Josip Peˇcari´c · Ksenija Smoljak

Received: 22 January 2013 / Revised: 3 May 2013 / Accepted: 7 May 2013 / Published online: 17 July 2014 © Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2014

Abstract In this paper, functionals defined as the difference between the left-hand and the right-hand sides of generalizations of Steffensen’s and Gauss’ inequality are studied. n-exponential and exponential as well as logarithmic convex functions related to defined functionals are produced. Furthermore, several families of exponentially convex functions are investigated, and new Stolarsky-type means related to defined functionals are generated. Keywords Steffensen’s inequality · Gauss’ inequality · Monotone function · Stolarsky-type means · Exponential convexity Mathematics Subject Classifications (2010) Primary 26D15 · Secondary 26D10

1 Introduction The well-known Steffensen inequality reads as follows [8, p. 181]: Theorem 1 Suppose that f is decreasing and g is integrable on [a, b] with 0 ≤ g ≤ 1 and b λ = a g(t)dt. Then we have 



b

f (t)dt≤ b−λ

b



a+λ

f (t)g(t)dt ≤

a

f (t)dt. a

The inequalities are reversed for f increasing.

J. Peˇcari´c · K. Smoljak () Faculty of Textile Technology, University of Zagreb, Prilaz baruna Filipovi´ca 28a, 10000 Zagreb, Croatia e-mail: [email protected] J. Peˇcari´c e-mail: [email protected]

(1)

ˇ ´ K. SMOLJAK J. PECARI C,

348

Let us recall the inequality of Gauss (see, [8, p. 195]): If f : [0, ∞) → R is a decreasing function, then, for all real numbers k > 0,   ∞ 4 ∞ 2 k2 f (x)dx ≤ x f (x)dx. 9 0 k

(2)

We continue this introduction by recalling theorems needed for defining linear functionals which will be studied in this paper. The following generalization of Steffensen’s inequality is given in [8, p. 192]: Theorem 2 Let h be a positive integrable function on [a, b] and f be an integrable function such that f (x)/ h(x) is increasing on [a, b]. If g is a real-valued integrable function such that 0 ≤ g(x) ≤ 1 for every x ∈ [a, b], then  b  a+λ f (t)g(t)dt ≥ f (t)dt, (3) a

a

where λ is the solution of the equation  b  a+λ h(t)dt = h(t)g(t)dt. a

(4)

a

If f (x)/ h(x) is a decreasing function, then the reverse of the inequality in (3) holds. Let conditions of Theorem 2 be fulfilled. Let linear functional L1 : C 1 [a, b] → R be defined by  b  a+λ L1 (f ) = f (t)g(t)dt − f (t)dt. (5) a

a

Remark 1 L1 (f ) ≥ 0 for all increasing functions f/ h. In 1982, Peˇcari´c et al. proved the following theorem which includes as special cases three famous inequalities: Volkov’s, Steffensen’s, and Ostrowski’s inequality (see [8, p. 194-195]): Theorem 3 Let G : [a, b] → R be an increasing and differentiable function and f : I → R be a decreasing function (I is an interval in R such that a, b, G(a), G(b) ∈ I ). (a)

If G(x) ≥ x, then



b



a

(b)



G(b)

f (x)G (x)dx ≥

f (x)dx.

(6)

G(a)

If G(x) ≤ x, then the reverse of the inequality in (6) is valid. The inequalities are reversed for f increasing.

Let the conditions of Theorem 3 be fulfilled. Let the linear functional L2 : C 1 (I ) → R be defined by   G(b) b  f (x)dx − a f (x)G (x)dx, G(x) ≥ x; G(a)  G(b) (7) L2 (f ) =  b  G(x) ≤ x. a f (x)G (x)dx − G(a) f (x)dx, Remark 2 L2 (f ) ≥ 0 for all increasing functions f .

STOLARKY-TYPE MEANS

349

In 1991, Alzer proved that an application of the following theorem leads to a new proof and to a converse of inequality (2) (see [1]): Theorem 4 Let g : [a, b] → R be strictly increasing, convex and differentiable, and let f : I → R be decreasing. Then  g(b)  b  b f (s(x))g (x)dx ≤ f (x)dx ≤ f (t (x))g  (x)dx, (8) a

g(a)

where s(x) =

a

g(b) − g(a) (x − a) + g(a) b−a

(9)

and t (x) = g  (x0 )(x − x0 ) + g(x0 ),

x0 ∈ [a, b].

(10)

(I ⊆ R is an interval containing a, b, g(a), g(b), t (a) and t (b)). If either g is concave (instead of convex) or f is increasing, then the reversed inequalities hold. Let the conditions of Theorem 4 be fulfilled. Let linear functionals L3 , L4 : C 1 (I ) → R be defined by  b  g(b)  f (s(x))g (x)dx − f (x)dx (11) L3 (f ) = a

and

 L4 (f ) =

g(b) g(a)

g(a)



b

f (x)dx −

f (t (x))g  (x)dx.

(12)

a

Remark 3 L3 (f ) ≥ 0 and L4 (f ) ≥ 0 for all increasing functions f . Furthermore, we recall the Cauchy-type mean value theorems related to functionals Li , i = 1, 2, 3, 4 (for details, see [4, 5], and [9]). Theorem 5 Let g be a real-valued integrable function such that 0 ≤ g(x) ≤ 1 for every x ∈ [a, b]. Let h be a positive function on (a, b] and derivable on (a, b), F, H be derivable on (a, b) such that F (x)/ h(x), H (x)/ h(x) ∈ C 1 [a, b] and such that H  (x)h(x) − H (x)h (x) = 0 for every x ∈ [a, b]. Then there exists ξ ∈ (a, b) such that   F (ξ ) F  (ξ )h(ξ ) − F (ξ )h (ξ ) L1 (F ) h(ξ ) =  =  (13)  . L1 (H ) H (ξ )h(ξ ) − H (ξ )h (ξ ) H (ξ ) h(ξ )

Theorem 6 Let I be a compact interval such that a, b, G(a), G(b) ∈ I . For F, H ∈ C 1 (I ),  H (x) = 0 for every x ∈ I and G : [a, b] → R an increasing and differentiable function there exists ξ ∈ I such that F  (ξ ) L2 (F ) =  . L2 (H ) H (ξ ) Theorem 7 Let g : [a, b] → R be a strictly increasing, convex, and differentiable function, s be defined by (9), and t be defined by (10). Let I be a compact interval such that a, b, g(a),

ˇ ´ K. SMOLJAK J. PECARI C,

350

g(b), t (a), t (b) ∈ I . Let F, H ∈ C 1 (I ), H  (x) = 0 for every x ∈ I . Then there exists ξ ∈ I such that F  (ξ ) Li (F ) =  , i = 3, 4. (14) Li (H ) H (ξ ) In this paper, we study functionals defined as the difference between the left-hand and the right-hand sides of generalizations of Steffensen’s and Gauss’ inequality given by (3), (6), and (8). Furthermore, we prove n-exponential convexity and define new Stolarsky-type means. The paper is organized as follows. After this introduction, in Section 2, we prove that functions related to functionals defined in the introduction are n-exponentially and exponentially as well as logarithmic convex. Applying results from Section 2 to several families of exponentially convex functions, in Section 3, we obtain new Stolarsky-type means related to defined functionals and prove monotonicity property of new means. Let us recall some notions: log denotes the natural logarithm function, id is the identity function on the actual set, and dx denotes the Lebesgue measure on R. 2 n-exponential convexity and exponential convexity We begin this section by introducing some necessary notations and recalling some basic facts about n-exponentially convex and exponentially convex functions (see, e.g., [2, 6, 7]). Definition 1 A function ψ : I → R is n-exponentially convex in the Jensen sense on I if  n  xi + xj ≥0 (15) ξi ξj ψ 2 i,j =1

holds for all choices ξi ∈ R and xi ∈ I , i = 1, . . . , n. A function ψ : I → R is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I . Remark 4 n-exponentially convex function in the Jensen sense is k-exponentially convex in the Jensen sense for every k ∈ N, k ≤ n. Definition 2 A function ψ : I → R is exponentially convex in the Jensen sense on I if it is n-exponentially convex in the Jensen sense for all n ∈ N. A function ψ : I → R is exponentially convex if it is exponentially convex in the Jensen sense and continuous. Remark 5 A positive function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense. A positive function is log-convex if and only if it is 2-exponentially convex. Proposition 1 If f is a convex function on I and if x1 ≤ y1 , x2 ≤ y2 , x1 = x2 , y1 = y2 , then the following inequality is valid: f (x2 ) − f (x1 ) f (y2 ) − f (y1 ) ≤ . x2 − x1 y2 − y1 If the function f is concave, the inequality is reversed.

STOLARKY-TYPE MEANS

351

Definition 3 Let f be a real-valued function defined on [a, b]. First-order divided difference of f at points x0 , x1 in [a, b] is defined by  f (x1 )−f (x0 ) , x0 = x1 x1 −x0 (16) [x0 , x1 ; f ] = f  (x0 ), x0 = x1 provided that f  exists. Our first objective is to produce n-exponentially convex and exponentially convex functions applying functionals Li , i = 1, 2, 3, 4 on a given family with the same property. Functionals Li , i = 2, 3, 4 are nonnegative for increasing function f , so results concerning these functionals will be grouped together. Functional L1 will be considered separately since it is nonnegative for increasing function f/ h, so we have some differences in our results. In the sequel, J and K will be intervals in R. Theorem 8 Let h be a positive function and ϒ = {fp / h : p ∈ K} be a family of functions defined on J such that the function p → [x0 , x1 ; fp / h] is n-exponentially convex in the Jensen sense on K for mutually different points x0 , x1 ∈ J . Let L1 be the linear functional defined by (5). Then p → L1 (fp ) is a n-exponentially convex function in the Jensen sense on K. If the function p → L1 (fp ) is continuous on K, then it is n-exponentially convex on K. Proof For ξj ∈ R, pj ∈ K, j = 1, . . . , n and pj k = g(x) =

n 

pj +pk 2 ,

we define the function

ξj ξk fpjk (x).

(17)

j,k=1

Since p → [x0 , x1 ; fp / h] is n-exponentially convex in the Jensen sense, we have

n

 fpjk g = ≥ 0, ξj ξk x 0 , x 1 ; x0 , x1 ; h h j,k=1

which implies that g/ h is an increasing function on J . Therefore, from Remark 1, we have L1 (g) ≥ 0, so n 

ξj ξk L1 (fpjk ) ≥ 0

j,k=1

holds. Hence, we can conclude that the function p → L1 (fp ) is n-exponentially convex on K in the Jensen sense. If the function p → L1 (fp ) is also continuous on K, then p → L1 (fp ) is n-exponentially convex by definition. Theorem 9 Let ϒ = {fp : p ∈ K} be a family of functions defined on J such that the function p → [x0 , x1 ; fp ] is n-exponentially convex in the Jensen sense on K for mutually different points x0 , x1 ∈ J . Let Li , i = 2, 3, 4 be linear functionals defined by (7), (11), and (12). Then p → Li (fp ) is a n-exponentially convex function in the Jensen sense on K. If the function p → Li (fp ) is continuous on K, then it is a n-exponentially convex on K.

ˇ ´ K. SMOLJAK J. PECARI C,

352

Proof As in proof of Theorem 8, we define a function g by (17). Since p → [x0 , x1 ; fp ] is n-exponentially convex in the Jensen sense, we have [x0 , x1 ; g] =

n 

ξj ξk [x0 , x1 ; fpjk ] ≥ 0,

j,k=1

which implies that g is an increasing function on J . Therefore, we have Li (g) ≥ 0, i = 2, 3, 4. Similar reasoning as in the proof of Theorem 8 completes the proof. The following corollary is a simple consequence of Theorem 8. Corollary 1 Let h be a positive function and ϒ = {fp / h : p ∈ K} be a family of functions defined on J such that the function p → [x0 , x1 ; fp / h] is exponentially convex in the Jensen sense on K for mutually different points x0 , x1 ∈ J . Let L1 be the linear functional defined by (5). Then p → L1 (fp ) is exponentially convex function in the Jensen sense on K. If the function p → L1 (fp ) is continuous on K, then it is an exponentially convex on K. The following corollary is a simple consequence of Theorem 9. Corollary 2 Let ϒ = {fp : p ∈ K} be a family of functions defined on J such that the function p → [x0 , x1 ; fp ] is exponentially convex in the Jensen sense on K for mutually different points x0 , x1 ∈ J . Let Li , i = 2, 3, 4 be linear functionals defined by (7), (11), and (12). Then p → Li (fp ) is an exponentially convex function in the Jensen sense on K. If the function p → Li (fp ) is continuous on K, then it is exponentially convex on K. Now, we will prove the corollaries of Theorems 8 and 9 which will be used in Section 3 for obtaining new Stolarsky type means. Corollary 3 Let h be a positive function and  = {fp / h : p ∈ K} be a family of functions defined on J such that the function p → [x0 , x1 ; fp / h] is 2-exponentially convex in the Jensen sense on K for mutually different points x0 , x1 ∈ J . Let L1 be the linear functional defined by (5). Then the following statements hold: (i) (ii)

If the function p → L1 (fp ) is positive and continuous on K, then it is 2-exponentially convex on K, and thus log-convex. If the function p → L1 (fp ) is strictly positive and differentiable on K, then for every p, q, u, v ∈ K such that p ≤ u and q ≤ v, we have Mp,q (L1 , ) ≤ Mu,v (L1 , ), where

⎧  1 ⎪ ⎨ L1 (fp ) p−q , L1 (f ) q d Mp,q (L1 , ) = L (f ) ⎪ ⎩ exp dp 1 p , L1 (fp )

(18)

p = q; p=q

for fp / h, fq / h ∈ . Proof (i)

This statement is a consequence of Theorem 8 and Remark 5.

(19)

STOLARKY-TYPE MEANS

(ii)

353

By (i) we have that p → L1 (fp ) is log-convex on K, that is, p → log L1 (fp ) is convex on K. Applying Proposition 1, we get log L1 (fp ) − log L1 (fq ) log L1 (fu ) − log L1 (fv ) ≤ p−q u−v

(20)

for p ≤ u, q ≤ v, p = q, u = v. Hence, we conclude that Mp,q (L1 , ) ≤ Mu,v (L1 , ). Cases p = q and u = v follow from (20) as limit cases. Corollary 4 Let  = {fp : p ∈ K} be a family of functions defined on J such that the function p → [x0 , x1 ; fp ] is 2-exponentially convex in the Jensen sense on K for mutually different points x0 , x1 ∈ J . Let Li , i = 2, 3, 4 be linear functionals defined by (7), (11), and (12). Then the following statements hold: (i) (ii)

If the function p → Li (fp ) is positive and continuous on K, then it is 2-exponentially convex on K, and thus log-convex. If the function p → Li (fp ) is strictly positive and differentiable on K, then for every p, q, u, v ∈ K such that p ≤ u and q ≤ v, we have Mp,q (Li , ) ≤ Mu,v (Li , ),

i = 2, 3, 4,

(21)

where Mp,q (Li , ) is defined by (19) for fp , fq ∈ . Proof Similar to the proof of Corollary 3. Remark 6 Results from Theorems 8 and 9 and Corollaries 1–4 still hold when x0 = x1 ∈ J for a family of differentiable functions with the same property. This follows from Definition 3.

3 Stolarsky-type means In this section, we will apply the general results obtained in the previous section to several families of functions. Using these families of functions, we will obtain new Stolarskytype means related to functionals defined in Section 1. We will give an explicit shape of the resulting inequalities in Stolarsky-type means only for functional L1 and for other functionals, it can be obtained in a similar way from a given general functional notation. In the sequel, I = [a, b] when we consider functional L1 , while I is as in Theorem 6 when we consider functional L2 , and as in Theorem 7 when we consider functionals L3 and L4 . Example 1 Let h be a positive integrable function and let 1 = {fp / h : (0, ∞) → R : p ∈ R} be a family of functions where fp is defined by  p x h(x), fp (x) = p log x h(x),

p = 0; p = 0.

(22)

ˇ ´ K. SMOLJAK J. PECARI C,

354

d fp (x) p−1 > 0 for x > 0, f / h is an increasing function for x > 0 and p dx h(x) = x d fp (x) p → dx h(x) is exponentially convex by definition. Similarly as in the proof of Theorem 8, we have that p → [x0 , x1 ; fp / h] is exponentially convex (and so exponentially convex in

Since

the Jensen sense). Using Corollary 1, we conclude that p → L1 (fp ) is exponentially convex in the Jensen sense. It is easy to verify that this mapping is continuous, so it is exponentially convex. Hence, we can apply Corollary 3 for this family of functions. From (19), we have 1  p−q  b  a+λ q a t p h(t)g(t)dt − a t p h(t)dt Mp,q (L1 , 1 ) = for p = q. (23)   p b t q h(t)g(t)dt − a+λ t q h(t)dt a

a

We can extend Mp,q (L1 , 1 ) to excluded cases:  b   a+λ p p t log t h(t)dt 1 a t log t h(t)g(t)dt − a Mp,p (L1 , 1 ) = exp − b  a+λ p p t p h(t)dt a t h(t)g(t)dt − a and

for p = 0

  b  a+λ 1 a log2 t h(t)g(t)dt − a log2 t h(t)dt . M0,0 (L1 , 1 ) = exp   2 b log t h(t)g(t)dt − a+λ log t h(t)dt a

(24) (25)

a

From (18), it follows that the function Mp,q (L1 , 1 ) is monotonic in parameters p and q. Remark 7 Let

1 = {fp : (0, ∞) → R : p ∈ R}

be a family of functions where fp is defined by  p x , fp (x) = p log x,

p = 0; p = 0.

Similarly as in Example 1, we obtain Mp,q (Li , 1 ), i = 2, 3, 4 defined by ⎧  1 Li (fp ) p−q ⎪ ⎪ , p = q; ⎪ L (f ) ⎪ i q  ⎨ Li (fp f0 ) 1  Mp,q (Li , 1 ) = exp Li (fp ) − p , p = q = 0;  ⎪ ⎪ ⎪ ⎪ exp Li (f02 ) , ⎩ p = q = 0. 2Li (f0 )

(26)

(27)

Theorem 5 applied to the functions fp / h, fq / h ∈ 1 and functional L1 and Theorems 6 and 7 applied to the functions fp , fq ∈ 1 and the functionals Li , i = 2, 3, 4 imply that there exists ξ ∈ I such that ξ p−q =

Li (fp ) , Li (fq )

i = 1, 2, 3, 4.

Since the function ξ → ξ p−q is invertible for p = q, we have 1  Li (fp ) p−q ≤ max I, i = 1, 2, 3, 4, min I ≤ Li (fq ) which together with the fact that Mp,q (L1 , 1 ) and Mp,q (Li , 1 ) are continuous, symmetric, and monotonic shows that Mp,q (L1 , 1 ) and Mp,q (Li , 1 ) are means.

STOLARKY-TYPE MEANS

355

Remark 8 Mean Mp,q (L1 , 1 ) is already obtained in papers [3] and [4] by using a different reasoning from the one mentioned above. Means Mp,q (Li , 1 ), i = 2, 3, 4, are obtained in [5] and [9]. Instead of the functional shape given by (27), their explicit shape can be found in the mentioned papers. Let us show that by using the same reasoning as in Example 1 for some other families of functions, we can obtain new means. Example 2 Let h be a positive integrable function and let 2 = {gp / h : R → (0, ∞) : p ∈ R} be a family of functions where gp is defined by  px e p h(x), gp (x) = x h(x),

p = 0; p = 0.

d gp (x) px > 0, g / h is an increasing function on R for every p ∈ R p dx h(x) = e d gp (x)

→ dx h(x) is exponentially convex by definition. As in Example 1, we conclude

→ L1 (gp ) is exponentially convex. For this family of functions, from (19), we have

(28)

Since

and

p p

that

1  p−q b  a+λ q a ept h(t)g(t)dt − a ept h(t)dt Mp,q (L1 , 2 ) = for p = q; (29)   p b eqt h(t)g(t)dt − a+λ eqt h(t)dt a a  b   a+λ pt pt e th(t)dt 1 a e th(t)g(t)dt − a Mp,p (L1 , 2 ) = exp  b for p = 0; (30) −  a+λ pt p ept h(t)dt a e h(t)g(t)dt − a  b   a+λ 1 a t 2 h(t)g(t)dt − a t 2 h(t)dt . (31) M0,0 (L1 , 2 ) = exp   2 b th(t)g(t)dt − a+λ th(t)dt



a

a

From (18), it follows that Mp,q (Li , 2 ) is monotonic in parameters p and q. Remark 9 Let

2 = {gp : R → (0, ∞) : p ∈ R}

be a family of functions where gp is defined by  px e p , gp (x) = x,

p = 0; p = 0.

Arguing similarly as in Example 2, we obtain Mp,q (Li , 2 ), i = 2, 3, 4, ⎧  1 Li (gp ) p−q ⎪ ⎪ , p = q; ⎪ ⎪ q )  ⎨ Li (g Li (gp ·g0 ) 1  Mp,q (Li , 2 ) = exp Li (gp ) − p , p = q = 0;  ⎪ ⎪ 2 ⎪ ⎪ ⎩ exp Li (g0 ) , p = q = 0. 2Li (g0 )

(32)

(33)

Applying Theorem 5 to the functions gp / h, gq / h ∈ 2 and functional L1 and Theorems 6 and 7 to the functions gp , gq ∈ 2 and functionals Li , i = 2, 3, 4, we get Sp,q (L1 , 2 ) = log Mp,q (L1 , 2 )

and Sp,q (Li , 2 ) = log Mp,q (Li , 2 )

ˇ ´ K. SMOLJAK J. PECARI C,

356

with min I ≤ Sp,q (L1 , 2 ) ≤ max I and min I ≤ Sp,q (Li , 2 ) ≤ max I . So Sp,q (L1 , 2 ) and Sp,q (Li , 2 ) are monotonic means by (18) and (21). Example 3 Let h be a positive integrable function and let 3 = {φp / h : (0, ∞) → (0, ∞) : p ∈ (0, ∞)} be a family of functions where φp is defined by  −x −p h(x), φp (x) = log p xh(x),

p = 1; p = 1.

(34)

d φp (x) −x > 0 for p, x ∈ (0, ∞), φ / h is an increasing function for x > 0. p dx h(x) = p d φp (x) d φp (x) −x is Laplace transform of nonnegative function, so p → dx dx h(x) = p h(x) is exponentially convex. As in Example 1, we conclude that p → L1 (φp ) is exponentially convex. For

Since

this family of functions, from (19), we have 1   p−q b  a+λ log q a p−t h(t)g(t)dt − a p−t h(t)dt Mp,q (L1 , 3 ) =   log p b q −t h(t)g(t)dt − a+λ q −t h(t)dt a

for p = q.

(35)

a

Moreover, by (19), we can extend Mp,q (L1 , 1 ) to excluded cases:  b   a+λ −1 a tp−t h(t)g(t)dt − a tp−t h(t)dt 1 Mp,p (L1 , 3 ) = exp − , p = 1   p b p−t h(t)g(t)dt − a+λ p−t h(t)dt p log p a a (36) and   b  a+λ −1 a t 2 h(t)g(t)dt − a t 2 h(t)dt . (37) M1,1 (L1 , 3 ) = exp b  a+λ 2 th(t)g(t)dt − th(t)dt a

a

Remark 10 Let 3 = {φp : (0, ∞) → (0, ∞) : p ∈ (0, ∞)} be a family of functions where φp is defined by  −x φp (x) =

−p log p

x,

,

p = 1; p = 1.

Similarly as in Example 3, we obtain that Mp,q (Li , 3 ), i = 2, 3, 4 defined by ⎧  1 Li (φp ) p−q ⎪ ⎪ , p = q; ⎪ ⎪  q ) ⎨ Li (φ −Li (φ1 ·φp ) 1 Mp,q (Li , 3 ) = exp p Li (φp ) − p log p , p = q = 1;  ⎪ ⎪ 2 ⎪ ⎪ ⎩ exp −Li (φ1 ) , p = q = 1. 2Li (φ1 ) Again, from Theorems 2–4, it follows that Sp,q (L1 , 3 ) = −L(p, q) log Mp,q (L1 , 3 ) and Sp,q (Li , 3 ) = −L(p, q) log Mp,q (Li , 3 )

(38)

(39)

STOLARKY-TYPE MEANS

357

with min I ≤ Sp,q (L1 , 3 ) ≤ max I and min I ≤ Sp,q (Li , 3 ) ≤ max I , where L(p, q) is p−q  logarithmic mean defined by L(p, q) = log p−log q . So Sp,q (L1 , 3 ) and Sp,q (Li , 3 ) are means, and by (18) and (21), they are monotonic. Example 4 Let h be a positive integrable function and let 4 = {ψp / h : (0, ∞) → (0, ∞) : p ∈ (0, ∞)} be a family of functions where ψp is defined by √ p

−e−x ψp (x) = √ p d ψp (x) dx h(x) = √ d ψp (x) −x p dx h(x) = e

Since

e−x

√ p

h(x).

(40)

> 0, ψp / h is an increasing function for x > 0. Note that ψ (x)

d p is a Laplace transform of a nonnegative function, so p → dx h(x) is exponentially convex. As in Example 1, we conclude that p → L1 (ψp ) is exponentially convex. For this family of functions, from (19), we have

√ b √  1 √  a+λ q a e−t p h(t)g(t)dt − a e−t p h(t)dt p−q Mp,q (Li , 4 ) = √  b √ √  p e−t q h(t)g(t)dt − a+λ e−t q h(t)dt a

and

for p = q (41)

a



 √ √ b  a+λ −1 a te−t p h(t)g(t)dt − a te−t p h(t)dt 1 Mp,p (Li , 4 ) = exp √  b √ . (42) − √  −t p h(t)g(t)dt − a+λ e −t p h(t)dt 2 p 2p ae a

Remark 11 Let 4 = {ψp : (0, ∞) → (0, ∞) : p ∈ (0, ∞)} be a family of functions where ψp is defined by √ p

−e−x ψp (x) = √ p

.

Similar as in Example 4, we obtain Mp,q (Li , 4 ), i = 2, 3, 4, where ⎧   1 ⎪ ⎨ Li (ψp ) p−q , p = q;  L (ψ ) q i Mp,q (Li , 4 ) =   −L (id·ψ ) ⎪ p 1 ⎩ exp √ i p = q. 2 p L (ψp ) − 2p ,

(43)

(44)

i

As in the previous examples, from Theorems 2–4, it follows that √ √ Sp,q (L1 , 4 ) = −( p + q) log Mp,q (L1 , 4 ) and

√ √ Sp,q (Li , 4 ) = −( p + q) log Mp,q (Li , 4 )

with min I ≤ Sp,q (L1 , 4 ) ≤ max I and min I ≤ Sp,q (Li , 4 ) ≤ max I . So Sp,q (L1 , 4 ) and Sp,q (Li , 4 ) are means, and by (18) and (21), they are monotonic. Acknowledgements The research of the authors was supported by the Croatian Ministry of Science, Education and Sports, under the Research Grant 117-1170889-0888.

ˇ ´ K. SMOLJAK J. PECARI C,

358

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

Alzer, H.: On an inequality of Gauss. Rev. Mat. Univ. Complut. Madrid 4(2-3), 179–183 (1991) Bernstein, S.N.: Sur les fonctions absolument monotones. Acta. Math. 52, 1–66 (1929) Jakˇseti´c, J., Peˇcari´c, J.: Steffensen’s means. J. Math. Inequal. 2(4), 487–498 (2008) Kruli´c, K., Peˇcari´c, J., Smoljak, K.: Gauss-Steffensen’s means. Math. Balkanica (N. S.) 24(1–2), 177– 190 (2010) Kruli´c, K., Peˇcari´c, J., Smoljak, K.: New generalized Steffensen means. Collect. Math. 62(2), 139–150 (2011) Mitrinovi´c, D.S., Peˇcari´c, J.E., Fink, A.M.: Classical and New Inequalities in Analysis. Kluwer Academic, The Netherlands (1993) Mitrinovi´c, D.S., Peˇcari´c, J.E.: On some inequalities for monotone functions. Boll. Un. Mat. Ital. B (7) 5(2), 407–416 (1991) Peˇcari´c, J.E., Proschan, F., Tong, Y.L.: Convex Functions, Partial Orderings, and Statistical Applications. Academic, San Diego (1992) Peˇcari´c, J., Smoljak, K.: Note on an inequality of Gauss. J. Math. Inequal. 5(2), 199–211 (2011)