Exponential convexity of Petrović and related functional

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.

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

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.

1. (i)

   f is exponentially convex.

2. (ii)

   f is continuous and

   \sum _{i,j=1}^n{\xi }_i{\xi }_{j}f\left(\frac{x_i+x_j}{2}\right)\ge 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

is positive semi-definite. In particular,

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)

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

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

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 {\sum }_{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.

1. (i)

   If there exists an m(≤ n) such that

   0\le {\bar{P}}_1\le {\bar{P}}_2\le \cdots \le {\bar{P}}_m\le 1,{\bar{P}}_{m+1}=\cdots ={\bar{P}}_n=0,

   (4)

whereP_k={\sum }_{i=1}^k{p}_i,{\bar{P}}_k=P_n-P_{k-1}\left(k=2,\dots ,n\right)and{\bar{P}}_1=P_n, then (3) holds.

1. (ii)

   If there exists an m(≤ n) such that

   {\bar{P}}_1\ge {\bar{P}}_2\ge \cdots \ge {\bar{P}}_m\ge 1,{\bar{P}}_{m+1}=\cdots ={\bar{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

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

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

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 {\sum }_{i=1}^n{p}_i\ne 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

(a) If{\int }_a^{b}h\left(t\right)\mathsf{\text{d}}g\left(t\right)\in Iand

then we have

(b) If{\int }_a^{b}h\left(t\right)\mathsf{\text{d}}g\left(t\right)\in Iand either

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

or

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

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.

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:

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

provided that f' exists.

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

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

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

(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

(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

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

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

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

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

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

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

If f(u)/u is strictly increasing for u ∈ I and all x _i 's are not equal or{\sum }_1^n{p}_i\ne 1then strict inequality holds in the above expression.

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

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

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{\mathcal{P}}_6\left(f\right).

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

if and only if, the relation

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^{\prime } 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^{\prime } 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^{\prime } 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 {\mathcal{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 functionst\mapsto {\mathcal{P}}_k\left(f_t\right)are log-convex in J-sense on (c, d).

(b) If the functionst\mapsto {\mathcal{P}}_k\left(f_t\right)are continuous on (c, d), then the functionst\mapsto {\mathcal{P}}_k\left(f_t\right)are log-convex on (c, d).

(c) If the functionst\mapsto {\mathcal{P}}_k\left(f_t\right)are derivable on (c, d), then for t, r, u, v ∈ (c, d) such that t ≤ u, r ≤ v, we have

where

Proof. (a) First, we prove log-convexity in J-sense of t\mapsto {\mathcal{P}}_k\left(f_t\right) 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

This gives

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

this implies

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, {\mathcal{P}}_k\left(f_t\right) are positive valued, and hence, by Lemma 2.4, the inequality (16) implies log-convexity in J-sense of the functions t\mapsto {\mathcal{P}}_k\left(f_t\right)for k = 1, 2, 3, 4.

Now we prove log-convexity in J-sense of t\mapsto {\mathcal{P}}_5\left(f_t\right). 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 {\mathcal{P}}_5\left(f_t\right) by using Remark 2.2 and Lemma 2.4.

At last, we prove log-convexity in J-sense of t\mapsto {\mathcal{P}}_k\left(f_t\right) for k = 6, 7, 8. For this, we consider the families of Functions defined in D _i for i = 5, 6, 7.

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

This gives

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

this implies

Since {\mathcal{P}}_k\left(f_t\right) are positive valued, we have by Lemma 2.4 and inequality (17) the log-convexity in J-sense of the functions t\mapsto {\mathcal{P}}_k\left(f_t\right) for k = 6, 7, 8.

(b) If t\mapsto {\mathcal{P}}_k\left(f_t\right) 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 {\mathcal{P}}_k\left(f_t\right) 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],

concluding

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

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^{\prime } 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^{\prime } 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^{\prime } 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{\mathcal{P}}_kbe 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) Ift\mapsto {\mathcal{P}}_k\left(f_t\right)are continuous on (c, d), then the functionst\mapsto {\mathcal{P}}_k\left(f_t\right), are exponentially convex on (c, d).

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

are positive semi-definite. In particular

(c) Ift\mapsto {\mathcal{P}}_k\left(f_t\right)are positive derivable on (c, d), then for t, r, u, v ∈ (c, d) such that t ≤ u, r ≤ v, we have

where{ℭ}_{k,i}\left(t,r;f_t\right)is defined similarly as in (15).

Proof. (a) First, we prove exponential convexity of t\mapsto {\mathcal{P}}_k\left(f_t\right) 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

This gives

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

thus

Hence t\mapsto {\mathcal{P}}_k\left(f_t\right) is exponentially convex for k = 1, 2, 3, 4.

Now we prove exponential convexity of t\mapsto {\mathcal{P}}_5\left(f_t\right). 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 {\mathcal{P}}_5\left(f_t\right) by using Remark 2.2.

At last, we prove exponential convexity of t\mapsto {\mathcal{P}}_k\left(f_t\right) for k = 6, 7, 8. For this, we consider the families of functions defined in E _i for i = 5, 6, 7.

For any n ∈ ℕ, ξ _i ∈ ℝ and t _i ∈ (c, d), i = 1, ..., n, we define

This gives

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

thus

Hence t\mapsto {\mathcal{P}}_k\left(f_t\right) are exponentially convex for k = 6, 7, 8.

(b) It follows by Proposition 1.2.

(c) Since t\mapsto {\mathcal{P}}_k\left(f_t\right) 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.

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

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 {\mathcal{P}}_k\left({\phi }_t\right) 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}\left(t,r;{\phi }_t\right)=A_{t,r}^1\left(\mathbf{x};\mathbf{p}\right) for t, r ≠ 1. By making substitution x_i\mapsto x_i^s, t ↦ t/s, r ↦ r/s and s ≠ 0, t, r ≠ s, we get {ℭ}_{1,2}\left(t,r;{\phi }_t\right)=A_{t,r}^s\left(\mathbf{x};\mathbf{p}\right) for t, r ≠ s, where A_{t,r}^s\left(\mathbf{x};\mathbf{p}\right) is defined in [5].

Similarly, {ℭ}_{4,2}\left(t,r;{\phi }_t\right)=C_{t,r}^1\left(\mathbf{x}\right)for t, r ≠ 1, and by substitution used above {ℭ}_{4,2}\left(t,r;{\phi }_t\right)=C_{t,r}^s\left(\mathbf{x}\right) for t, r ≠ s, where C_{t,r}^s\left(\mathbf{x}\right) is defined in [6].

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

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 {\mathcal{P}}_5\left({\beta }_t\right), which have been proved in [9].

Since {\beta }_t^{\prime }\left(u\right)=u^{t-1}={\mathsf{\text{e}}}^{\left(t-1\right)logu}, the mapping t\mapsto {\beta }_t^{\prime } 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

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 {\mathcal{P}}_k\left({\delta }_t\right) for k = 6, 7, 8, which have been proved in [10].

Since {\delta }_t^{″}\left(u\right)=u^{t-2}={\mathsf{\text{e}}}^{\left(t-2\right)logu}, the mapping t\mapsto {\delta }_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}\left(t,r;{\delta }_t\right)=B_{t,r}^1\left(\mathbf{x};\mathbf{p}\right) for t, r ≠ 1. By making substitution x_i\mapsto x_i^s, t ↦ t/s, r ↦ r/s and s ≠ 0, t, r ≠ s, we get {ℭ}_{6,7}\left(t,r;{\delta }_t\right)=B_{t,r}^s\left(\mathbf{x};\mathbf{p}\right) for t, r ≠ s, where B_{t,r}^s\left(\mathbf{x};\mathbf{p}\right) is defined in [10].

Similarly, {ℭ}_{7,7}\left(t,r;{\delta }_t\right)=F_{t,r}^1\left(a,b,h,g\right) for t, r ≠ 1 and by substitution used above {ℭ}_{7,7}\left(t,r;{\delta }_t\right)=F_{t,r}^s\left(a,b,h,g\right) for t, r ≠ s, where F_{t,r}^s\left(a,b,h,g\right) is defined in [6].

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

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 {\mathcal{P}}_k\left({\zeta }_t\right) 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 {\mathcal{P}}_k\left({\zeta }_t\right) for k = 1, 2, 3, 4.

For {\mathcal{P}}_1\left(f_t\right) using the function ζ _t in Theorem 2.6, ℭ_1,2 (t, r; ζ _t ) in this particular case looks like

where {\tilde{x}}_n={\sum }_{i=1}^n{p}_i{x}_i.

For {\mathcal{P}}_4\left(f_t\right) using the function ζ _t in Theorem 2.6, ℭ_4,2 (t, r; ζ _t ) in this particular case looks like

where {\widehat{x}}_n=\left(x_1-{\sum }_{i=2}^n{x}_i\right).

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

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 {\mathcal{P}}_5\left({\theta }_t\right).

Since {\theta }_t^{\prime }\left(u\right)=t^{-u}, the mapping t\mapsto {\theta }_t^{\prime }\left(u\right) is exponentially convex function [7]. If we choose f _t = θ _t in Theorem 2.6, we get exponential convexity of the functional {\mathcal{P}}_5\left({\theta }_t\right).