Exponential convexity of Petrović and related functional

Saad I Butt; Josip Pečarić; Atiq Ur Rehman

doi:10.1186/1029-242X-2011-89

Research
Open Access
Published: 20 October 2011

Exponential convexity of Petrović and related functional

Saad I Butt¹,
Josip Pečarić^1,2 &
Atiq Ur Rehman³

Journal of Inequalities and Applications volume 2011, Article number: 89 (2011) Cite this article

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Abstract

We consider functionals due to the difference in Petrović and related inequalities and prove the log-convexity and exponential convexity of these functionals by using different families of functions. We construct positive semi-definite matrices generated by these functionals and give some related results. At the end, we give some examples.

1 Introduction

First time exponentially convex functions are introduced by Bernstein [1]. Independently of Bernstein, but some what later Widder [2] introduced these functions, as a sub-class of convex functions in a given interval (a, b), and denoted this class by W_a,b. After the initial development, there is a big gap in time before applications and examples of interest were constructed. One of the reasons is that, aside from absolutely monotone functions and completely monotone functions, as special classes of exponentially convex functions, there is no operative criteria to recognize exponential convexity of functions.

Definition 1. [[3], p. 373] A function f : (a, b) → ℝ is exponentially convex if it is continuous and

∑ i, j = 1 n ξ i ξ j f (x i + x j) ≥ 0

(1)

for all n ∈ ℕ and all choices ξ_i ∈ ℝ and x_i + x_j ∈ (a, b), 1 ≤ i, j ≤ n.

Proposition 1.1. Let f : (a, b) → ℝ. The following propositions are equivalent.

(i)
f is exponentially convex.
(ii)
f is continuous and
$∑ i, j = 1 n ξ i ξ j f x i + x j 2 ≥ 0$

for every ξ_i ∈ ℝ and every x_i ∈ (a, b), 1 ≤ i ≤ n.

Proposition 1.2. If f is exponentially convex, then the matrix

f x i + x j 2 i, j = 1 n

is positive semi-definite. In particular,

det f x i + x j 2 i, j = 1 n ≥ 0

for every n ∈ ℕ, x_i ∈ (a, b), i = 1, ..., n.

Proposition 1.3. If f : (a, b) → (0, ∞) is an exponentially convex function, then f is log-convex which means that for every x, y ∈ (a, b) and all λ ∈(0, 1)

f (λ x + (1 - λ) y) ≤ f (x) λ f (y) 1 - λ .

We consider functionals due to the differences in the Petrović and related inequalities. These inequalities are given in the following theorems [[4], pp. 152-159].

Theorem 1.4. Let I = (0, a] ⊆ ℝ be an interval, (x₁, ..., x_n ) ∈ Iⁿ, and (p₁, ..., p_n ) be a non-negative n-tuple such that

∑ i = 1 n p i x i ∈ I a n d ∑ i = 1 n p i x i ≥ x j f o r j = 1, …, n .

(2)

If f : I → ℝ be a function such that f (x)/x is an increasing for x ∈ I, then

f ∑ i = 1 n p i x i ≥ ∑ i = 1 n p i f (x i) .

(3)

Remark 1.5. Let us note that if f(x)/x is a strictly increasing function for x ∈ I, then equality in (3) is valid if we have equalities in (2) instead of the inequalities, that is, x₁= ⋯ = x_n and $∑ i = 1 n p i = 1$ .

Theorem 1.6. Let I = (0, a] ⊆ ℝ be an interval, (x₁, ..., x_n ) ∈ Iⁿ, such that 0 < x₁ ≤ ⋯ ≤ x_n , (p₁, ..., p_n ) be a non-negative n-tuple and f : I → ℝ be a function such that f(x)/x is an increasing for x ∈ I.

(i)
If there exists an m(≤ n) such that
$0 ≤ P ̄ 1 ≤ P ̄ 2 ≤ ⋯ ≤ P ̄ m ≤ 1, P ̄ m + 1 = ⋯ = P ̄ n = 0,$
(4)

where $P k = ∑ i = 1 k p i, P ̄ k = P n - P k - 1 (k = 2, …, n)$ and $P ̄ 1 = P n$ , then (3) holds.

(ii)
If there exists an m(≤ n) such that
$P ̄ 1 ≥ P ̄ 2 ≥ ⋯ ≥ P ̄ m ≥ 1, P ̄ m + 1 = ⋯ = P ̄ n = 0,$
(5)

then the reverse of inequality in (3) holds.

Theorem 1.7. Let I = (0, a] ⊆ ℝ be an interval, (x₁, ..., x_n ) ∈ Iⁿ, and x₁ - x₂ - ... - x_n ∈ I. Also let f : I → ℝ be a function such that f(x)/x is an increasing for x ∈ I. Then

f x 1 - ∑ i = 2 n x i ≤ f (x 1) - ∑ i = 2 n f (x i) .

(6)

Remark 1.8. If f(x)/x is a strictly increasing function for x ∈ I, then strict inequality holds in (6).

Theorem 1.9. Let I = (0, a] ⊆ ℝ be an interval, (x₁, ..., x_n ) ∈ Iⁿ , (p₁, ..., p_n ) and (q₁, ..., q_n ) be non-negative n-tuples such that (2) holds. If f : I → ℝ be an increasing function, then

∑ i = 1 n q i f ∑ i = 1 n p i x i ≥ ∑ i = 1 n q i f (x i) .

(7)

Remark 1.10. If f is a strictly increasing function on I and all x_i's are not equal, then we obtain strict inequality in (7).

Theorem 1.11. Let I = [0, a] ⊆ ℝ be an interval, (x₁, ..., x_n ) ∈ Iⁿ, and (p₁, ..., p_n ) be a non-negative n-tuple such that (2) holds.

If f is a convex function on I, then

f ∑ i = 1 n p i x i ≥ ∑ i = 1 n p i f (x i) + 1 - ∑ i = 1 n p i f (0) .

(8)

Remark 1.12. In the above theorem, if f is a strictly convex, then inequality in (8) is strict, if all x_i's are not equal or $∑ i = 1 n p i ≠ 1$ .

Theorem 1.13. Let I ⊆ ℝ be an interval, 0 ∈ I, f be a convex function on I, h : [a.b] → I be continuous and monotonic with h(t₀) = 0, t₀ ∈ [a, b] be fixed, g be a function of bounded variation and

G (t) : = ∫ a t d g (x), G ¯ (t) : = ∫ t b d g (x) .

(a) If $∫ a b h (t) d g (t) ∈ I$ and

0 ≤ G (t) ≤ 1 f o r a ≤ t ≤ t 0, 0 ≤ G ¯ (t) ≤ 1 f o r t 0 < t ≤ b,

(9)

then we have

∫ a b f (h (t)) d g (t) ≥ f ∫ a b h (t) d g (t) + ∫ a b d g (t) - 1 f (0) .

(10)

(b) If $∫ a b h (t) d g (t) ∈ I$ and either

there exists an s ≤ t₀such that G(t) ≤ 0 for t < s,

G (t) ≥ 1 f o r s ≤ t ≤ t 0 a n d G ¯ (t) ≤ 0 f o r t > t 0

(11)

or

there exists an s ≥ t₀such that G(t) ≤ 0 for t < t₀,

G ¯ (t) ≥ 1 f o r t 0 < t < s, a n d G ¯ (t) ≤ 0 f o r t ≥ s,

(12)

then the reverse of the inequality in (10) holds.

In this paper, we consider certain families of functions to prove log-convexity and exponential convexity of functionals due to the differences in inequalities given in Theorems 1.4-1.13. We construct positive semi-definite matrices generated by these functionals. Also by using log-convexity of these functionals, we prove monotonicity of the expressions introduced by these functionals. At the end, we give some examples.

2 Main results

Let I ⊆ ℝ be an interval and f : I → ℝ be a function. Then for distinct points u_i ∈ I, i = 0, 1, 2, the divided differences in first and second order are defined as follows:

u i, u i + 1, f = f (u i + 1) - f (u i) u i + 1 - u i i = 0, 1, u 0, u 1, u 2, f = u 1, u 2, f - u 0, u 1, f u 2 - u 0 .

(13)

The values of the divided differences are independent of the order of the points u₀, u₁, u₂ and may be extended to include the cases when some or all points are equal, that is

[u 0, u 0, f] = lim u 1 → u 0 [u 0, u 1, f] = f ′ (u 0),

provided that f' exists.

Now passing through the limit u₁ → u₀ and replacing u₂ by u in (13), we have [[4], p. 16]

[u 0, u 0, u, f] = lim u 1 → u 0 [u 0, u 1, u, f] = f (u) - f (u 0) - (u - u 0) f ′ (u 0) (u - u 0) 2, u ≠ u 0,

provided that f' exists. Also passing to the limit u_i → u (i = 0, 1, 2) in (13), we have

[u, u, u, f] = lim u i → u [u 0, u 1, u 2, f] = f ″ (u) 2,

provided that f″ exists.

One can note that if for all u₀, u₁ ∈ I, [u₀, u₁, f] ≥ 0, then f is increasing on I and if for all u₀, u₁, u₂ ∈ I, [u₀, u₁, u₂, f] ≥ 0, then f is convex on I.

(M₁) Under the assumptions of Theorem 1.4, with all x_i 's not equal, we define a linear functional as

P 1 (f) = f ∑ i = 1 n p i x i - ∑ i = 1 n p i f (x i) .

(M₂) Under the assumptions of Theorem 1.6, with all x_i 's not equal and (4) is valid, we define a linear functional as

P 2 (f) = P 1 (f) .

(M₃) Under the assumptions of Theorem 1.6, with all x_i 's not equal and (5) is valid, we define a linear functional as

P 3 (f) = - P 1 (f) .

(M₄) Under the assumptions of Theorem 1.7, with all x_i 's not equal, we define a linear functional as

P 4 (f) = f (x 1) - ∑ i = 2 n f (x i) - f x 1 - ∑ i = 2 n x i .

(M₅) Under the assumptions of Theorem 1.9, with all x_i 's not equal, we define a linear functional as

P 5 (f) = ∑ i = 1 n q i f ∑ i = 1 n p i x i - ∑ i = 1 n q i f (x i) .

(M₆) Under the assumptions of Theorem 1.11, with all x_i 's not equal, we define a linear functional as

P 6 (f) = f ∑ i = 1 n p i x i - ∑ i = 1 n p i f (x i) - 1 - ∑ i = 1 n p i f (0) .

(M₇) Under the assumptions of Theorem 1.13, such that (9) is valid, we define a linear functional as

P 7 (f) = ∫ a b f (h (t)) d g (t) - f ∫ a b h (t) d g (t) - ∫ a b d g (t) - 1 f (0) .

(M₈) Under the assumptions of Theorem 1.13, such that (11) or (12) is valid, we define a linear functional as

P 8 (f) = - P 7 (f) .

Remark 2.1. Under the assumptions of (M_k ) for k = 1, 2, 3, 4, if f(u)/u is an increasing function for u ∈ I, then

P k (f) ≥ 0, f o r k = 1, 2, 3, 4 .

If f(u)/u is strictly increasing for u ∈ I and all x_i's are not equal or $∑ 1 n p i ≠ 1$ then strict inequality holds in the above expression.

Remark 2.2. Under the assumptions of (M₅), if f is an increasing function on I, then

P 5 (f) ≥ 0 .

If f is strictly increasing function on I and all x_i's are not equal, then we obtain strict inequality in the above expression.

Remark 2.3. Under the assumptions of (M_k ) for k = 6, 7, 8, if f is a convex function on I, then

P k (f) ≥ 0 f o r k = 6, 7, 8 .

If f is strictly increasing function on I and all x_i's are not equal, then we obtain strict inequality in the above expression for $P 6 (f)$ .

The following lemma is nothing more than the discriminant test for the non-negativity of second-order polynomials.

Lemma 2.4. Let I ⊆ ℝ be an interval. A function f : I → (0, ∞) is log-convex in J-sense on I, that is, for each r, t ∈ I

f (r) f (t) ≥ f 2 t + r 2

if and only if, the relation

m 2 f (t) + 2 m n f t + r 2 + n 2 f (r) ≥ 0

(14)

holds for each m, n ∈ ℝ and r, t ∈ I.

To define different families of functions, let I ⊆ ℝ and (c, d) ⊆ ℝ be intervals. For distinct points u₀, u₁, u₂ ∈ I we suppose

D₁ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₁, F_t ] is log-convex in J-sense, where F_t (u) = f_t (u)/u}.

D₂ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, F_t ] is log-convex in J-sense, where F_t (u) = f_t (u)/u and $F t ′$ exists}.

D₃ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₁, f_t ] is log-convex in J-sense}.

D₄ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, f_t ] is log-convex in J-sense, where $f t ′$ exists}.

D₅ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₁, u₂, f_t ] is log-convex in J-sense}.

D₆ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, u₂, f_t ] is log-convex in J-sense, where $f t ′$ exists}.

D₇ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, u₀, f_t ] is log-convex in J-sense, where $f t ″$ exists}.

In this theorem, we prove log-convexity in J-sense, log-convexity and related results of the functionals associated with their respective families of functions.

Theorem 2.5. Let $P k$ be the linear functionals defined in(M_k ), associate the functionals with D_i in such a way that, for k = 1, 2, 3, 4, f_t ∈ D _i, i = 1, 2, for k = 5, f_t ∈ D _i, i = 3, 4 and for k = 6, 7, 8, f_t ∈ D _i, i = 5, 6, 7. Also for k = 7, 8, assume that the linear functionals are positive. Then, the following statements are valid:

(a) The functions $t ↦ P k (f t)$ are log-convex in J-sense on (c, d).

(b) If the functions $t ↦ P k (f t)$ are continuous on (c, d), then the functions $t ↦ P k (f t)$ are log-convex on (c, d).

(c) If the functions $t ↦ P k (f t)$ are derivable on (c, d), then for t, r, u, v ∈ (c, d) such that t ≤ u, r ≤ v, we have

B k, i (t, r; f t) ≤ B k, i (u, v; f t),

where

B k, i (t, r; f t) = P k (f t) P k (f r) 1 t - r, t ≠ r, exp d d t (P k (f t)) P k (f t), t = r .

(15)

Proof. (a) First, we prove log-convexity in J-sense of $t ↦ P k (f t)$ for k = 1, 2, 3, 4. For this, we consider the families of functions defined in D₁ and D₂.

Choose any m, n ∈ ℝ, and t, r ∈ (c, d), we define the function

h (u) = m 2 f t (u) + 2 m n f t + r 2 (u) + n 2 f r (u) .

This gives

u 0, u 1, H = m 2 u 0, u 1, F t + 2 m n u 0, u 1, F t + r 2 + n 2 u 0, u 1, F r,

where H (u) = h(u)/u and F_t (u) = f_t (u)/u.

Since t ↦ [u₀, u₁, F_t ] is log-convex in J-sense, by Lemma 2.4 the right-hand side of above expression is non-negative. This implies h(u)/u is an increasing function for u ∈ I.

Thus by Remark 2.1

P k (h) ≥ 0 for k = 1, 2, 3, 4,

this implies

m 2 P k (f t) + 2 m n P k (f t + r 2) + n 2 P k (f r) ≥ 0 .

(16)

Now [u₀, u₁, F_t ] > 0 as it is log-convex, this implies f_t (u)/u is strictly increasing for all u ∈ I and t ∈ (c, d). Also all x_i 's are not equal and therefore by Remark 2.1, $P k (f t)$ are positive valued, and hence, by Lemma 2.4, the inequality (16) implies log-convexity in J-sense of the functions $t ↦ P k (f t)$ for k = 1, 2, 3, 4.

Now we prove log-convexity in J-sense of $t ↦ P 5 (f t)$ . For this, we consider the families of functions defined in D₃ and D₄. Following the same steps as above and having H(u) = h(u), we have the log-convexity in J-sense of $P 5 (f t)$ by using Remark 2.2 and Lemma 2.4.

At last, we prove log-convexity in J-sense of $t ↦ P k (f t)$ for k = 6, 7, 8. For this, we consider the families of Functions defined in D_ifor i = 5, 6, 7.

Choose any m, n ∈ ℝ, and t, r ∈ (c, d), we define the function

h (u) = m 2 f t (u) + 2 m n f t + r 2 (u) + n 2 f r (u) .

This gives

u 0, u 1, u 2, h = m 2 u 0, u 1, u 2, f t + 2 m n u 0, u 1, u 2, f t + r 2 + n 2 u 0, u 1, u 2, f r .

Since t ↦ [u₀, u₁, u₂, f_t ] is log-convex in J-sense, by Lemma 2.4 the right-hand side of above expression is non-negative. This implies h is a strictly convex function on I.

Thus by Remark 2.3

P k (h) ≥ 0 for k = 6, 7, 8,

this implies

m 2 P k (f t) + 2 m n P k (f t + r 2) + n 2 P k (f r) ≥ 0 .

(17)

Since $P k (f t)$ are positive valued, we have by Lemma 2.4 and inequality (17) the log-convexity in J-sense of the functions $t ↦ P k (f t)$ for k = 6, 7, 8.

(b) If $t ↦ P k (f t)$ are additionally continuous for k = 1, ..., 8 and D _i's associated with them, then these are log-convex, since J-convex continuous functions are convex functions.

(c) Since the functions log $P k (f t)$ are convex for k = 1, ..., 8, and D _i's associated with them, therefore for t ≤ u, r ≤ v, t ≠ r, u ≠ v, we have [[4], p.2],

log P k (f t) - log P k (f r) t - r ≤ log P k (f u) - log P k (f v) u - v,

concluding

B k, i (t, r; f t) ≤ B k, i (u, v; f t) .

Now if t = r ≤ u, we apply lim_r→t, concluding,

B k, i (t, t; f t) ≤ B k, i (u, v; f t) .

Other possible cases are treated similarly.

In order to define different families of functions related to exponential convexity, let I ⊆ ℝ and (c, d) ⊆ ℝ be any intervals. For distinct points u₀, u₁, u₂ ∈ I we suppose

E₁ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₁, F_t ] is exponentially convex, where F_t (u) = f_t (u)/u}.

E₂ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, F_t ] is exponentially convex, where F_t (u) = f_t (u)/u and $F t ′$ exists}.

E₃ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₁, f_t ] is exponentially convex}.

E₄ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, f_t ] is exponentially convex, where $f t ′$ exists}.

E₅ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₁, u₂, f_t ] is exponentially convex}.

E₆ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, u₂, f_t ] is exponentially convex, where $f t ′$ exists}.

E₇ = {f_t : I → ℝ | t ∈ (c, d) and t ↦ [u₀, u₀, u₀, f_t ] is exponentially convex, where $f t ″$ exists}.

In this theorem, we prove the exponential convexity of the functionals associated with their respective families of functions. Also we define positive semi-definite matrices for these functionals and give some related results.

Theorem 2.6. Let $P k$ be the linear functionals defined in (M_k ), associate the functionals with E_i in such a way that, for k = 1, 2, 3, 4, f_t ∈ E _i, i = 1, 2, for k = 5, f_t ∈ E _i, i = 3, 4 and for k = 6, 7, 8, f_t ∈ E _i, i = 5, 6, 7. Then, the following statements are valid:

(a) If $t ↦ P k (f t)$ are continuous on (c, d), then the functions $t ↦ P k (f t)$ , are exponentially convex on (c, d).

(b) For every q ∈ℕ and t₁, ..., t_q ∈ (c, d), the matrices

P k (f t l + t m 2) l, m = 1 q

are positive semi-definite. In particular

det P k (f t l + t m 2) l, m = 1 s ≥ 0 f o r s = 1, 2, …, q .

(c) If $t ↦ P k (f t)$ are positive derivable on (c, d), then for t, r, u, v ∈ (c, d) such that t ≤ u, r ≤ v, we have

ℭ k, i (t, r; f t) ≤ ℭ k, i (u, v; f t)

where $ℭ k, i (t, r; f t)$ is defined similarly as in (15).

Proof. (a) First, we prove exponential convexity of $t ↦ P k (f t)$ for k = 1, 2, 3, 4. For this, we consider the families of functions defined in E₁ and E₂.

For any n ∈ ℕ, ξ_i ∈ ℝ and t_i ∈ (c, d), i = 1, ..., n, we define

h (u) = ∑ i, j = 1 n ξ i ξ j f t i + t j 2 (u) .

This gives

u 0, u 1, H = ∑ i, j = 1 n ξ i ξ j u 0, u 1, F t i + t j 2,

where H (u) = h(u)/u and F_t (u) = f_t (u)/u.

Since t ↦ [u₀, u₁, F_t ] is exponentially convex, right-hand side of the above expression is non-negative, which implies h(u)/u is an increasing function on I.

Thus by Remark 2.1, we have

P k (h) ≥ 0, for k = 1, 2, 3, 4,

thus

∑ i, j = 1 n ξ i ξ j P k f t i + t j 2 ≥ 0 .

Hence $t ↦ P k (f t)$ is exponentially convex for k = 1, 2, 3, 4.

Now we prove exponential convexity of $t ↦ P 5 (f t)$ . For this, we consider the families of functions defined in E₃ and E₄. Following the same steps as above and having H (u) = h(u), we have the exponential convexity of the $P 5 (f t)$ by using Remark 2.2.

At last, we prove exponential convexity of $t ↦ P k (f t)$ for k = 6, 7, 8. For this, we consider the families of functions defined in E_ifor i = 5, 6, 7.

For any n ∈ ℕ, ξ_i ∈ ℝ and t_i ∈ (c, d), i = 1, ..., n, we define

h (u) = ∑ i, j = 1 n ξ i ξ j f t i + t j 2 (u) .

This gives

u 0, u 1, u 2, h = ∑ i, j = 1 n ξ i ξ j u 0, u 1, u 2, f t i + t j 2 .

Since t ↦ [u₀, u₁, u₂, f_t ] is exponentially convex therefore right-hand side of the above expression is non-negative, which implies h(u) is a strictly convex function on I.

Thus by Remark 2.3, we have

P k (h) ≥ 0 for k = 6, 7, 8,

thus

∑ i, j = 1 n ξ i ξ j P k f t i + t j 2 ≥ 0 .

Hence $t ↦ P k (f t)$ are exponentially convex for k = 6, 7, 8.

(b) It follows by Proposition 1.2.

(c) Since $t ↦ P k (f t)$ are positive derivable for k = 1, ..., 8 with E _i's associated with them, we have our conclusion using part (c) of the Theorem 2.5.

3 Examples

In this section, we will vary on choices of families of functions in order to construct different examples of log and exponentially convex functions and related results.

Example 1. Let t ∈ ℝ and φ_t : (0, ∞) → ℝ be the function defined as

φ t (u) = u t t - 1, t ≠ 1, u log u, t = 1 .

(18)

Then φ_t (u)/u is strictly increasing on (0, ∞) for each t ∈ ℝ. One can note that t ↦ [u₀, u₀, φ_t (u)/u] is log-convex for all t ∈ ℝ. If we choose f_t = φ_t in Theorem 2.5, we get log-convexity of the functionals $P k (φ t)$ for k = 1, 2, 3, 4, which have been proved in [5, 6].

Since φ_t (u)/u)' = u^t-2= e^{(t - 2) log u}, the mapping t ↦ (φ_t (u)/u)' is exponentially convex [7]. If we choose f_t = φ_t in Theorem 2.6, we get results that have been proved in [6, 8]. Also we get $ℭ 1, 2 (t, r; φ t) = A t, r 1 (x; p)$ for t, r ≠ 1. By making substitution $x i ↦ x i s$ , t ↦ t/s, r ↦ r/s and s ≠ 0, t, r ≠ s, we get $ℭ 1, 2 (t, r; φ t) = A t, r s (x; p)$ for t, r ≠ s, where $A t, r s (x; p)$ is defined in [5].

Similarly, $ℭ 4, 2 (t, r; φ t) = C t, r 1 (x)$ for t, r ≠ 1, and by substitution used above $ℭ 4, 2 (t, r; φ t) = C t, r s (x)$ for t, r ≠ s, where $C t, r s (x)$ is defined in [6].

Example 2. Let t ∈ ℝ and β_t : (0, ∞) → ℝ be the function defined as

β t (u) = u t t, t ≠ 0, log u, t = 0 .

(19)

Then, β_t is strictly increasing on (0, ∞) for each t ∈ ℝ. One can note that t ↦ [u₀, u₀, β_t ] is log-convex for all t ∈ ℝ. If we choose f_t = β_t in Theorem 2.5, we get log-convexity of the functional $P 5 (β t)$ , which have been proved in [9].

Since $β t ′ (u) = u t - 1 = e (t - 1) log u$ , the mapping $t ↦ β t ′$ is exponentially convex [7]. If we choose f_t = β_t in Theorem 2.6, we get results that have been proved in [9]. Also we get ℭ_5,4 (t, r; β_t ) = H_t,r (x; p; q) for t, r ≠ 0, where H_t,r (x; p; q) is defined in [9].

Example 3. Let t ∈ (0, ∞) and δ_t : [0, ∞) → ℝ be the function defined as

δ t (u) = u t t (t - 1), t ≠ 1, u log u, t = 1,

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with a convention that 0 log 0 = 0. Then δ_t is convex on [0, ∞) for each t ∈ (0, ∞). One can note that t ↦ [u₀, u₀, u₀, δ_t ] is log-convex for all t ∈ (0, ∞). If we choose f_t = δ_t in Theorem 2.5, we get log-convexity of the functionals $P k (δ t)$ for k = 6, 7, 8, which have been proved in [10].

Since $δ t ″ (u) = u t - 2 = e (t - 2) log u$ , the mapping $t ↦ δ t ″$ is exponentially convex [7]. If we choose f_t = δ_t in Theorem 2.6, we get results that have been proved in [8, 10]. Also we get $ℭ 6, 7 (t, r; δ t) = B t, r 1 (x; p)$ for t, r ≠ 1. By making substitution $x i ↦ x i s$ , t ↦ t/s, r ↦ r/s and s ≠ 0, t, r ≠ s, we get $ℭ 6, 7 (t, r; δ t) = B t, r s (x; p)$ for t, r ≠ s, where $B t, r s (x; p)$ is defined in [10].

Similarly, $ℭ 7, 7 (t, r; δ t) = F t, r 1 (a, b, h, g)$ for t, r ≠ 1 and by substitution used above $ℭ 7, 7 (t, r; δ t) = F t, r s (a, b, h, g)$ for t, r ≠ s, where $F t, r s (a, b, h, g)$ is defined in [6].

Example 4. Let t ∈ (0, ∞) and ζ_t : (0, ∞) → ℝ be the function defined as

ζ t (u) = u t - u - log t, t ≠ 1, u 2, t = 1 .

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One can note that t ↦ [u₀, u₀, ζ_t (u)/u] is log-convex for all t ∈ (0, ∞). If we choose f_t = ζ_t in Theorem 2.5, we get log-convexity of the functionals $P k (ζ t)$ for k = 1, 2, 3, 4.

Since ζ_t (u)/u)' = t^-u, the mapping t ↦ (ζ_t (u)/u)' is exponentially convex [7]. If we choose f_t = ζ_t in Theorem 2.6, we get exponential convexity of the functionals $P k (ζ t)$ for k = 1, 2, 3, 4.

For $P 1 (f t)$ using the function ζ_t in Theorem 2.6, ℭ_1,2 (t, r; ζ_t ) in this particular case looks like

ℭ 1, 2 (t, r; ζ t) = log r x ̃ n t - x ̃ n - ∑ i = 1 n p i x i t - x i log t x ̃ n r - x ̃ n - ∑ i = 1 n p i x i r - x i 1 t - r, t ≠ r, t, r ≠ 1, x ̃ n t - x ̃ n - ∑ i = 1 n p i x i t - x i - log t x ̃ n 2 - ∑ i = 1 n p i x i 2 1 t - 1, t ≠ r = 1, exp - 1 t log t - x ̃ n 2 t - x ̃ n - ∑ i = 1 n p i x i 2 t - x i t x ̃ n t - x ̃ n - ∑ i = 1 n p i x i t - x i, t = r, t, r ≠ 1, exp x ̃ n 3 - ∑ i = 1 n p i x i 3 - 2 x ̃ n 2 - ∑ i = 1 n p i x i 2, t = r = 1,

where $x ̃ n = ∑ i = 1 n p i x i$ .

For $P 4 (f t)$ using the function ζ_t in Theorem 2.6, ℭ_4,2 (t, r; ζ_t ) in this particular case looks like

ℭ 4, 2 (t, r; ζ t) = log r x 1 t - x 1 - ∑ i = 2 n x i t - x i - x^n t - x^n log t x 1 r - x 1 - ∑ i = 2 n x i r - x i - x^n r - x^n 1 t - r, t ≠ r, t, r ≠ 1, x 1 t - x 1 - ∑ i = 2 n x i t - x i - x^n t - x^n - log t x 12 - ∑ i = 2 n x i 2 - x^n 2 1 t - 1, t ≠ r = 1, exp - 1 t log t - x 12 t - x 1 - ∑ i = 2 n x i 2 t - x i - x^n 2 t - x^n t x 1 t - x 1 - ∑ i = 2 n x i t - x i - x^n t - x^n, t = r, t, r ≠ 1, exp x 13 - ∑ i = 2 n x i 3 - x^n 3 - 2 x 12 - ∑ i = 2 n x i 2 - x^n 2, t = r = 1,

where $x^n = (x 1 - ∑ i = 2 n x i)$ .

Example 5. Let t ∈ (0, ∞) and θ_t : (0, ∞) → ℝ be the function defined as

θ t (u) = t - u - log t, t ≠ 1, u, t = 1 .

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One can note that t ↦ [u₀, u₀, θ_t ] is log-convex for all t ∈ (0, ∞), and if we choose f_t = θ_t in Theorem 2.5, we get log-convexity of the functional $P 5 (θ t)$ .

Since $θ t ′ (u) = t - u$ , the mapping $t ↦ θ t ′ (u)$ is exponentially convex function [7]. If we choose f_t = θ_t in Theorem 2.6, we get exponential convexity of the functional $P 5 (θ t)$ .

For $P 5 (f t)$ using the function θ_t in Theorem 2.6, ℭ_5,4 (t, r; θ_t ) in this particular case looks like

ℭ 5, 4 (t, r; θ t) = log r ∑ i = 1 n q i t - x ̃ n - ∑ i = 1 n q i t - x i log t ∑ i = 1 n q i r - x ̃ n - ∑ i = 1 n q i r - x i 1 t - r, t ≠ r, t, r ≠ 1, ∑ i = 1 n q i t - x ̃ n - ∑ i = 1 n q i t - x i - log t ∑ i = 1 n q i x ̃ n - ∑ i = 1 n q i x i 1 t - 1, t ≠ r = 1, exp - 1 t log t - ∑ i = 1 n q i x ̃ n t - x ̃ n - ∑ i = 1 n q i x i t - x i - t ∑ i = 1 n q i t - x ̃ n - ∑ i = 1 n q i t - x i, t = r, t, r ≠ 1, exp ∑ i = 1 n q i x ̃ n 2 - ∑ i = 1 n q i x i 2 - 2 ∑ i = 1 n q i x ̃ n - ∑ i = 1 n q i x i, t = r = 1,

Where $x ̃ n = ∑ i = 1 n p i x i$ .

Example 6. Let t ∈ (0, ∞) and λ_t : (0, ∞) → ℝ be the function defined as

λ t (u) = u e - u t - t .

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One can note that t ↦ [u₀, u₀, λ_t (u)/u] is log-convex for all t ∈ (0, ∞). If we choose f_t = λ_t in Theorem 2.5, we get log-convexity of the functionals $P k (λ t)$ for k = 1, 2, 3, 4.

Since $λ t (u) ∕ u ′ = e - u t$ , the mapping t ↦ (λ_t (u)/u)' is exponentially convex function [7]. If we choose f_t = λ_t in Theorem 2.6, we get exponential convexity of the functionals $P k (λ t)$ for k = 1, 2, 3, 4.

For $P 1 (f t)$ using the function λ_t in Theorem 2.6, ℭ_1,2 (t, r; λ_t ) in this particular case looks like

ℭ 1, 2 (t, r; λ t) = r x ̃ n e - x ̃ n t - ∑ i = 1 n p i x i e - x i t t x ̃ n e - x ̃ n r - ∑ i = 1 n p i x i e - x i r 1 t - r, t ≠ r, exp - 1 2 t - x ̃ n 2 e - x ̃ n t - ∑ i = 1 n p i x i 2 e - x i t 2 t x ̃ n e - x ̃ n t - ∑ i = 1 n p i x i e - x i t, t = r,

Where $x ̃ n = ∑ i = 1 n p i x i$ .

Now for $P 4 (f t)$ using the function λ_t in Theorem 2.6, ℭ_4,2 (t, r; λ_t ) in this particular case looks like

ℭ 4, 2 (t, r; λ t) = r x 1 e - x 1 t - ∑ i = 2 n x i e - x i t - x^n e - x^n t t x 1 e - x 1 t - ∑ i = 2 n x i e - x i r - x^n e - x^n r 1 t - r, t ≠ r, exp - 1 2 t - x 12 e - x 1 t - ∑ i = 2 n x i 2 e - x i t - x^n 2 e - x^n t 2 t x 1 e - x 1 t - ∑ i = 2 n x i e - x i t - x^n e - x^n t, t = r,

Where $x^n = (x 1 - ∑ i = 2 n x i)$ .

Example 7. Let t ∈ (0, ∞) and ξ_t : (0, ∞)→ ℝ, be the function defined as

ξ t (u) = e - u t - t .

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One can note that t ↦ [u₀, u₀, ξ_t ] is log-convex for all t ∈ (0, ∞). If we choose f_t = ξ_t in Theorem 2.5, we get log-convexity of the functional $P 5 (ξ t)$ .

Since $ξ t ′ (u) = e - u t$ , the mapping $t ↦ ξ t ′ (u)$ is exponentially convex function [7]. If we choose f_t = ξ_t in Theorem 2.6 we get exponential convexity of the functional $P 5 (ξ t)$ .

For $P 5 (f t)$ using the function ξ_t in Theorem 2.6, ℭ_5,4 (t, r; ξ_t ) in this particular case looks like

ℭ 5, 4 (t, r; ξ t) = r ∑ i = 1 n q i e - x ̃ n t - ∑ i = 1 n q i e - x i t t ∑ i = 1 n q i e - x ̃ n r - ∑ i = 1 n q i e - x i r 1 t - r, t ≠ r, exp - 1 2 t - ∑ i = 1 n q i x ̃ n e - x ̃ n t - ∑ i = 1 n q i x i e - x i t 2 t ∑ i = 1 n q i e - x ̃ n t - ∑ i = 1 n q i e - x i t, t = r,

Where $x ̃ n = ∑ i = 1 n p i x i$ .

Example 8. Let t ∈ ℝ and ψ_t : (0, ∞) → ℝ be the function defined as

ψ t (u) = u e u t t, t ≠ 0, u 2, t = 0 .

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One can note that t ↦ [u₀, u₀, ψ_t (u)/u] is log-convex for all t ∈ ℝ. If we choose f_t = ψ_t in Theorem2.5, we get log-convexity of the functionals $P k (ψ t)$ for k = 1, 2, 3, 4.

Since (ψ_t (u)/u)' = e ^ut , the mapping t ↦ (ψ_t (u)/u)' is exponentially convex function [7]. If we choose f_t = ψ_t in Theorem 2.6 we get exponential convexity of the functionals $P k (ψ t)$ for k = 1, 2, 3, 4.

For $P 1 (f t)$ using the function ψ_t in Theorem 2.6, ℭ_1,2 (t, r; ψ_t ) in this particular case looks like

ℭ 1, 2 (t, r; ψ t) = r x ̃ n e x ̃ n t - ∑ i = 1 n p i x i e x i t t x ̃ n e x ̃ n r - ∑ i = 1 n p i x i e x i r 1 t - r, t ≠ r, t, r ≠ 0, x ̃ n e x ̃ n t - ∑ i = 1 n p i x i e x i t t x ̃ n 2 - ∑ i = 1 n p i x i 2 1 t - 1, t ≠ r = 0, exp - 1 t + x ̃ n 2 e x ̃ n t - ∑ i = 1 n p i x i 2 e x i t x ̃ n e x ̃ n t - ∑ i = 1 n p i x i e x i t, t = r, t, r ≠ 0, exp x ̃ n 3 - ∑ i = 1 n p i x i 3 2 x ̃ n 2 - ∑ i = 1 n p i x i 2, t = r = 0,

Where $x ̃ n = ∑ i = 1 n p i x i$ .

Now for $P 4 (f t)$ using the function ψ_t in Theorem 2.6, ℭ_4,2 (t, r; ψ_t ) in this particular case looks like

ℭ 4, 2 (t, r; ψ t) = r x 1 e x 1 t - ∑ i = 2 n x i e x i t - x^n e x^n t x 1 e x 1 r - ∑ i = 2 n x i e x i r - x^n e x^n r 1 t - r, t ≠ r, t, r ≠ 0, x 1 e x 1 t - ∑ i = 2 n x i e x i t - x^n e x^n t x 12 - ∑ i = 2 n x i 2 - x^n 2 1 t - 1, t ≠ r = 0, exp - 1 t + x 12 e x 1 t - ∑ i = 2 n x i 2 e x i t - x^n 2 e x^n t x 1 e x 1 t - ∑ i = 2 n x i e x i t - x^n e x^n t, t = r, t, r ≠ 0, exp x 13 - ∑ i = 2 n x i 3 - x^n 3 2 x 12 - ∑ i = 2 n x i 2 - x^n 2, t = r = 0,

where $x^n = (x 1 - ∑ i = 2 n x i)$ .

Example 9. Let t ∈ ℝ and ω_t : (0, ∞) → ℝ be the function defined as

ω t (u) = e u t t, t ≠ 0, u, t = 0 .

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One can note that t ↦ [u₀, u₀, ω_t ] is log-convex for all t ∈ ℝ. If we choose f_t = ω_t in Theorem 2.5, we get log-convexity of the functional $P 5 (ω t)$ .

Since $ω t ′ (u) = e u t$ , the mapping $t ↦ ω t ′ (u)$ is exponentially convex function [7]. If we choose f_t = ω_t in Theorem 2.6 we get exponential convexity of the functional $P 5 (ω t)$ .

For $P 5 (f t)$ using the function ω_t in Theorem 2.6, ℭ_5,4 (t, r; ω_t ) in this particular case looks like

ℭ 5, 4 (t, r; ω t) = r ∑ i = 1 n q i e x ̃ n t - ∑ i = 1 n q i e x i t t ∑ i = 1 n q i e x ̃ n r - ∑ i = 1 n q i e x i r 1 t - r, t ≠ r, t, r ≠ 0, ∑ i = 1 n q i e x ̃ n t - ∑ i = 1 n q i e x i t t ∑ i = 1 n q i x ̃ n - ∑ i = 1 n q i x i 1 t - 1, t ≠ r = 0, exp - 1 t + ∑ i = 1 n q i x ̃ n e x ̃ n t - ∑ i = 1 n q i x i e x i t ∑ i = 1 n q i e x ̃ n t - ∑ i = 1 n q i e x i t, t = r, t, r ≠ 0, exp ∑ i = 1 n q i x ̃ n 2 - ∑ i = 1 n q i x i 2 2 ∑ i = 1 n q i x ̃ n - ∑ i = 1 n q i x i, t = r = 0,

Where $x ̃ n = ∑ i = 1 n p i x i$ .

References

1.
Bernstein SN: Sur les fonctions absolument monotones. Acta Math 1929, 52: 1–66. 10.1007/BF02592679
Article Google Scholar
2.
Widder DV: The Laplace Transform. Princeton 1946, 1941.
Google Scholar
3.
Mitrinović DS, Pečarić J, Fink AM: Classical and New Inequalities in Analysis. Kluwer, The Netherlands; 1993.
Google Scholar
4.
Pečarić J, Proschan F, Tong YL: Convex Functions, Partial Orderings and Statistical Applications, vol. 187 of Mathematics in Science and Engineering. Academic Press, Boston; 1992.
Google Scholar
5.
Pečarić J, Ur Rehman A: On logarithmic convexity for power sums and related results. J Inequal Appl 2008, 2008: 9. Article ID 389410
Google Scholar
6.
Pečarić J, Farid G, Ur Rehman A: On refinements of Aczél, Popoviciu, Bellmans inequalities and related results. J Inequal Appl 2010, 2010: 17. Article ID 579567
Google Scholar
7.
Jakšetić J, Pečarić J: Exponential convexity method., in press.
8.
Anwar M, Jakšetić J, Pečarić J, Ur Rehman A: Exponential convexity, positive semi-definite matrices and fundamental inequalities. J Math Inequal 2010,4(2):171–189.
MathSciNet Article Google Scholar
9.
Pečarić J, Ur Rehman A: On exponentially convexity for power sums and related results. J Math Inequal (to appear)
10.
Pečarić J, Ur Rehman A: On logarithmic convexity for power sums and related results II. J Inequal Appl 2008, 2008: 12. Article ID 305623
Google Scholar

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Acknowledgements

The authors wish to thank the anonymous referees for their very careful reading of the manuscript and fruitful comments and suggestions. This research was partially funded by Higher Education Commission, Pakistan. The research of the first author was supported by the Croatian Ministry of Science, Education and Sports under the Research Grant 117-1170889-0888.

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Abdus Salam School of Mathematical Sciences, GC University, Lahore, Pakistan
Saad I Butt & Josip Pečarić
Faculty of Textile Technology, University of Zagreb, Zagreb, Croatia
Josip Pečarić
Department of Mathematics, University of Sargodha, Sargodha, Pakistan
Atiq Ur Rehman

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Saad I Butt
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Butt, S.I., Pečarić, J. & Ur Rehman, A. Exponential convexity of Petrović and related functional. J Inequal Appl 2011, 89 (2011). https://doi.org/10.1186/1029-242X-2011-89

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Received: 22 March 2011
Accepted: 20 October 2011
Published: 20 October 2011
DOI: https://doi.org/10.1186/1029-242X-2011-89

Keywords

convex functions
divided difference
exponentially convex
functionals
log-convex functions
positive semi-definite