# Superadditivity, Monotonicity, and Exponential Convexity of the Petrović-Type Functionals

Authors: Saad Ihsan Butt; Mario Krnić; Josip Pečarić
Journal: Abstract and Applied Analysis
 (2012)

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## Abstract

We consider functionals derived from Petrović-type inequalities and establish their superadditivity, subadditivity, and monotonicity properties on the corresponding realn-tuples. By virtue of established results we also define some related functionals and investigate their properties regarding exponential convexity. Finally, the general results are then applied to some particular settings.


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## Body

## 1. Introduction

In this paper we prove some interesting properties of the functionals derived by virtue of the Petrović and related inequalities (see, [1] pages 152–159). For the sake of simplicity these inequalities will be referred to as the Petrović-type inequalities, while the corresponding functionals will be referred to as the Petrović-type functionals.Therefore, throughout this introduction, we present the above-mentioned Petrović-type inequalities that will be the base in our research and also define the corresponding functionals that will be the subject of our study. We start with the following inequality.Theorem 1.1.
LetI=(0,a]⊆ℝ+ be an interval, (x1,…,xn)∈In, and let (p1,…,pn)∈ℝ+n be a nonnegative real n-tuple such that
(1.1)∑i=1npixi∈I,∑i=1npixi≥xjforj=1,…,n.
If f:I→ℝ is such that the function f(x)/x is decreasing on I, then
(1.2)f(∑i=1npixi)≤∑i=1npif(xi).
In addition, if f(x)/x is increasing on I, then the sign of inequality in (1.2) is reversed.Remark 1.2.
It should be noticed here that, iff(x)/x is strictly increasing function on I, then the equality in (1.2) is valid if and only if we have equalities in (1.1) instead of inequalities, that is, if x1=⋯=xn and ∑i=1npi=1.Motivated by the above theorem, we define the Petrović-type functional𝒫1, as a difference between the right-hand side and the left-hand side of inequality (1.2), that is,(1.3)P1(x,p;f)=∑i=1npif(xi)-f(∑i=1npixi),
where x=(x1,…,xn)∈In, I=(0,a], p=(p1,…,pn)∈ℝ+n, and f is defined on the interval I.Remark 1.3.
If (1.1) holds and f(x)/x is decreasing on I, then
(1.4)P1(x,p;f)≥0.
On the other hand, if (1.1) is valid and f(x)/x is increasing on I, then
(1.5)P1(x,p;f)≤0.
The above functional (1.3) will also be considered under slightly altered assumptions on real n-tuples x and p. For that sake, the following result from [1] will be used in due course.Theorem 1.4.
SupposeI=(0,a]⊆ℝ+, (x1,…,xn)∈In is a real n-tuple such that 0<x1≤⋯≤xn, and let (p1,…,pn)∈ℝ+n. Further, let f:I→ℝ be such that f(x)/x is increasing on I.(i)
If there existsm(≤n) such that
(1.6)P¯1≥P¯2≥⋯≥P¯m≥1,P¯m+1=⋯=P¯n=0,
where Pk=∑i=1kpi,P¯k=Pn-Pk-1,k=2,…,n, and P¯1=Pn, then (1.2) holds.(ii)
If there existsm(≤n) such that
(1.7)0≤P¯1≤P¯2≤⋯≤P¯m≤1,P¯m+1=⋯=P¯n=0,
then the reverse inequality in (1.2) holds.Remark 1.5.
Iff(x)/x is increasing on I and (1.6) holds, then the Petrović-type functional 𝒫1 is nonnegative, that is, inequality (1.4) is valid. Conversely, if f(x)/x is increasing on I and conditions as in (1.7) are fulfilled, then relation (1.5) holds.In order to define another Petrović-type functional, we cite the following Petrović-type inequality involving a convex function.Theorem 1.6.
LetI=[0,a]⊆ℝ+, (x1,…,xn)∈In and let (p1,…,pn)∈ℝ+n fulfill conditions as in (1.1). If f:I→ℝ is a convex function, then
(1.8)f(∑i=1npixi)≥∑i=1npif(xi)+(1-∑i=1npi)f(0).Remark 1.7.
Iff is a concave function then -f is convex, hence replacing f by -f in Theorem 1.6, we obtain inequality
(1.9)f(∑i=1npixi)≤∑i=1npif(xi)+(1-∑i=1npi)f(0).Remark 1.8.
If the functionf from Theorem 1.6 is strictly convex, then the inequality in (1.8) is strict, if all xi’s are not equal or ∑i=1npi≠1.Now, regarding inequality (1.8) we define another Petrović-type functional 𝒫2 by the formula(1.10)P2(x,p;f)=f(∑i=1npixi)-∑i=1npif(xi)-(1-∑i=1npi)f(0),
provided that x=(x1,…,xn)∈In, I=[0,a], p=(p1,…,pn)∈ℝ+n, and f is defined on I.Remark 1.9.
If (1.1) holds and f:I→ℝ is a convex function, then
(1.11)P2(x,p;f)≥0.
If (1.1) holds and f:I→ℝ is a concave function, then
(1.12)P2(x,p;f)≤0.
Finally, we will also be concerned with an integral form of the Petrović-type functional, based on the following integral Petrović-type inequality.Theorem 1.10.
LetI⊆ℝ be an interval, 0∈I, and let f:I→ℝ be a convex function. Further, suppose h:[a,b]→I is continuous and monotone with h(t0)=0, where t0∈[a,b] is fixed, and g is a function of bounded variation with
(1.13)G(t):=∫atdg(x),G¯(t):=∫tbdg(x).(a)
If∫abh(t)dg(t)∈I and
(1.14)0≤G(t)≤1fora≤t≤t0,0≤G¯(t)≤1fort0≤t≤b,
then
(1.15)∫abf(h(t))dg(t)≥f(∫abh(t)dg(t))+(∫abdg(t)-1)f(0).(b)
If∫abh(t)dg(t)∈I and either there exists an s≤t0 such that G(t)≤0 for t<s,
(1.16)G(t)≥1fors≤t≤t0,G¯(t)≤0fort>t0,
or there exists an s≥t0 such that G(t)≤0 for t<t0,
(1.17)G¯(t)≥1fort0<t<s,G¯(t)≤0fort≥s,
then the reverse inequality in (1.15) holds.In view of Theorem1.10, we define the functional(1.18)P3(h,g;f)=∫abf(h(t))dg(t)-f(∫abh(t)dg(t))-(∫abdg(t)-1)f(0),
which represents the integral form of the Petrović-type functional.Remark 1.11.
If the functionsf, g, and h are defined as in the statement of Theorem 1.10 and (1.14) holds, then the functional 𝒫3 is nonnegative, that is,
(1.19)P3(h,g;f)≥0.
Moreover, if either (1.16) or (1.17) holds then
(1.20)P3(h,g;f)≤0.
For a comprehensive inspection on the Petrović-type inequalities including proofs and diverse applications, the reader is referred to [1].
The paper is organized in the following way. After this introduction, in Section2 we prove superadditivity, subadditivity, and monotonicity properties of functionals 𝒫1,𝒫2, and 𝒫3. In addition, we also derive some bounds for the functional 𝒫1 via the nonweighted functional of the same type. By virtue of results from Section 2, in Section 3 we study some other classes of Petrović-type functionals and investigate their properties regarding exponential convexity. Finally, in Section 4 we apply our general results to some particular settings.Convention.
Throughout this paperℝ denotes the set of real numbers, while ℝ+ denotes the set of nonnegative numbers (including zero). Further, bold letters p, q, and x, respectively, denote real n-tuples (p1,p2,…,pn), (q1,q2,…,qn), and (x1,x2,…,xn). Moreover, p≥q means that pi≥qi for all i=1,2,…,n.


## 2. Main Results

In this section we derive some interesting properties of the Petrović-type functionals𝒫1,𝒫2, and 𝒫3, defined in Section 1. More precisely, we establish the conditions under which the appropriate functional is superadditive (subadditive) and increasing (decreasing), with respect to the corresponding n-tuple of real numbers. Our first result refers to the Petrović-type functional 𝒫1 defined by (1.3).Theorem 2.1.
LetI=(0,a]⊆ℝ+, x∈In, and let nonnegative n-tuples p, q fulfill conditions as in (1.1). If f:I→ℝ is such that the function f(x)/x is decreasing on I, then the functional (1.3) possess the following properties.(i)
𝒫1(x,.;f) is superadditive on nonnegative n-tuples, that is,
(2.1)P1(x,p+q;f)≥P1(x,p;f)+P1(x,q;f),
provided that ∑i=1n(pi+qi)xi∈I.(ii)
Ifp,q∈ℝ+n are such that p≥q and ∑i=1n(pi-qi)xi≥xj, j=1,…,n, then
(2.2)P1(x,p;f)≥P1(x,q;f)≥0,
that is, 𝒫1(x,.;f) is increasing on nonnegative n-tuples.(iii)
Iff(x)/x is increasing on I, then the signs of inequalities in (2.1) and (2.2) are reversed, that is, 𝒫1(x,.;f) is subadditive and decreasing on nonnegative n-tuples.Proof.
(i) Using definition (1.3) of the Petrović-type functional 𝒫1 and utilizing the linearity of the sum, we have
(2.3)P1(x,p+q;f)=∑i=1n(pi+qi)f(xi)-f(∑i=1n(pi+qi)xi)=∑i=1npif(xi)+∑i=1nqif(xi)-f(∑i=1npixi+∑i=1nqixi).
On the other hand, since f(x)/x is decreasing function, Theorem 1.1 in the nonweighted case (for n=2) yields inequality
(2.4)f(∑i=1npixi+∑i=1nqixi)≤f(∑i=1npixi)+f(∑i=1nqixi).
Finally, combining relations (2.3) and (2.4), we obtain
(2.5)P1(x,p+q;f)≥∑i=1npif(xi)+∑i=1nqif(xi)-f(∑i=1npixi)-f(∑i=1nqixi).
Therefore we have
(2.6)P1(x,p+q;f)≥P1(x,p;f)+P1(x,q;f),
as claimed.
(ii) Monotonicity follows easily from the superadditivity property. Sincep≥q≥0, we can represent p as the sum of two nonnegative n-tuples, namely, p=(p-q)+q. Now, from relation (2.1) we get
(2.7)P1(x,p;f)=P1(x,p-q+q;f)≥P1(x,p-q;f)+P1(x,q;f).
Finally, if the conditions as in (ii) are fulfilled, then, taking into account Theorem 1.1 we have that 𝒫1(x,p-q;f)≥0, which implies that 𝒫1(x,p;f)≥𝒫1(x,q;f).
(iii) The case of increasing functionf(x)/x is treated in the same way as in (i) and (ii), taking into account that the sign of the corresponding Petrović-type inequality is reversed.By virtue of Theorem1.4, the above properties of the functional 𝒫1 can also be derived in a slightly different setting.Theorem 2.2.
LetI=(0,a]⊆ℝ+, x∈In, and let real n-tuples p, q fulfill conditions as in (1.6). If f:I→ℝ is such that the function f(x)/x is increasing on I, then the functional 𝒫1 has the following properties.(i)
𝒫1(x,.;f) is superadditive on real n-tuples, that is,
(2.8)P1(x,p+q;f)≥P1(x,p;f)+P1(x,q;f),
provided that ∑i=1n(pi+qi)xi∈I and 0<∑i=1npixi≤∑i=1nqixi.(ii)
If0<x1≤⋯≤xn, p≥q, and there exist m(≤n) such that
(2.9)P¯1-Q¯1≥P¯2-Q¯2≥⋯≥P¯m-Q¯m≥1,P¯m+1=Q¯m+1=⋯=P¯n=Q¯n=0,
where Pk=∑i=1kpi,Qk=∑i=1kqi, P¯k-Q¯k=(Pn+Qn)-(Pk-1+Qk-1),k=2,…,n, P¯1=Pn, and Q¯1=Qn, then
(2.10)P1(x,p;f)≥P1(x,q;f)≥0,
that is, 𝒫1(x,.;f) is increasing on real n-tuples.(iii)
If realn-tuples p and q fulfill conditions as in (1.7), then the signs of inequalities in (2.8) and (2.10) are reversed, that is, 𝒫1(x,·;f) is subadditive and decreasing on real n-tuples.Proof.
(i) The proof follows the same lines as the proof of the previous theorem. Namely, the left-hand side of (2.8) can be rewritten as
(2.11)P1(x,p+q;f)=∑i=1n(pi+qi)f(xi)-f(∑i=1n(pi+qi)xi)=∑i=1npif(xi)+∑i=1nqif(xi)-f(∑i=1npixi+∑i=1nqixi).
Moreover, f(x)/x is increasing, hence Theorem 1.4 for n=2 yields inequality
(2.12)f(∑i=1npixi+∑i=1nqixi)≤f(∑i=1npixi)+f(∑i=1nqixi).
Finally, relations (2.11) and (2.12) imply inequality
(2.13)P1(x,p+q;f)≥∑i=1npif(xi)+∑i=1nqif(xi)-f(∑i=1npixi)-f(∑i=1nqixi),
that is, we obtain (2.8).
(ii) Consideringp≥q≥0, the real n-tuple p can be rewritten as p=(p-q)+q. Now, regarding relation (2.8) we have
(2.14)P1(x,p;f)=P1(x,p-q+q;f)≥P1(x,p-q;f)+P1(x,q;f).
Finally, taking into account conditions as in (2.9), it follows by Theorem 1.4 that 𝒫1(x,p-q;f)≥0, that is, 𝒫1(x,p;f)≥𝒫1(x,q;f), which completes the proof.
(iii) This case is treated in the same way as in (i) and (ii), taking into account that the sign of the corresponding Petrović-type inequality is reversed.Superadditivity and monotonicity properties stated in Theorem2.1 play an important role in numerous applications of the Petrović-type inequalities. In the sequel we utilize the monotonicity property of the Petrović-type functional 𝒫1. More precisely, we derive some bounds for this functional, expressed in terms of the nonweighted functional of the same type.Corollary 2.3.
LetI=(0,a]⊆ℝ+, x∈In, and let f:I→ℝ be such that f(x)/x is decreasing on I. Further, suppose p∈ℝ+n is such that ∑i=1n(pi-m)xi≥xj and ∑i=1n(M-pi)xi≥xj, j=1,2,…,n, where m=min1≤i≤n{pi} and M=max1≤i≤n{pi}.
Ifm>1 then the Petrović-type functional 𝒫1 fulfills inequality
(2.15)P1(x,p;f)≥mP10(x;f),
while for M<1 one has
(2.16)P1(x,p;f)≤MP10(x;f),
where
(2.17)P10(x;f)=∑i=1nf(xi)-f(∑i=1nxi).
Moreover, if f(x)/x is increasing on I, then the signs of inequalities in (2.15) and (2.16) are reversed.Proof.
Sincep=(p1,…,pn)≥m=(m,m,…,m), monotonicity of the Petrović-type functional implies that 𝒫1(x,p;f)≥𝒫1(x,m;f).
On the other hand, iff(x)/x is decreasing function, we have
(2.18)f(au)≤af(u),a>1,f(au)≥af(u),a<1.
Now, regarding (2.18) we have
(2.19)P1(x,m;f)=m∑i=1nf(xi)-f(m∑i=1nxi)≥m∑i=1nf(xi)-mf(∑i=1nxi),
that is, we obtain (2.15). Inequality (2.16) is derived in a similar way, by using the second inequality in (2.18).Our next result provides superadditivity and monotonicity properties of the Petrović-type functional defined by (1.10).Theorem 2.4.
LetI=[0,a]⊆ℝ+, x∈In, and let p,q∈ℝ+n fulfill conditions as in (1.1). If f:I→ℝ is a convex function, then the functional (1.10) has the following properties:(i)
𝒫2(x,·;f) is superadditive on nonnegative n-tuples, that is,
(2.20)P2(x,p+q;f)≥P2(x,p;f)+P2(x,q;f),
provided that ∑i=1n(pi+qi)xi∈I.(ii)
Ifp,q are such that p≥q and ∑i=1n(pi-qi)xi≥xj,j=1,…,n, then
(2.21)P2(x,p;f)≥P2(x,q;f)≥0,
that is, 𝒫2(x,.;f) is increasing on nonnegative n-tuples.(iii)
Iff:I→ℝ is a concave function, then the signs of inequalities in (2.20) and (2.21) are reversed, that is, 𝒫2(x,.;f) is subadditive and decreasing on nonnegative n-tuples.Proof.
(i) The left-hand side of inequality (2.20) can be rewritten as
(2.22)P2(x,p+q;f)=f(∑i=1n(pi+qi)xi)-∑i=1n(pi+qi)f(xi)-(1-∑i=1n(pi+qi))f(0)=f(∑i=1npixi+∑i=1nqixi)-∑i=1npif(xi)-∑i=1nqif(xi)-(1-(∑i=1npi+∑i=1nqi))f(0).
Further, Theorem 1.6 in the nonweighted case (for n=2) yields inequality
(2.23)f(∑i=1npixi+∑i=1nqixi)≥f(∑i=1npixi)+f(∑i=1nqixi)-f(0),
hence combining relations (2.22) and (2.23), we get
(2.24)P2(x,p+q;f)≥f(∑i=1npixi)-∑i=1npif(xi)-(1-∑i=1npi)f(0)+f(∑i=1npixi)-∑i=1nqif(xi)-(1-∑i=1nqi)f(0).
Thus, considering definition (1.10) we obtain (2.20), as claimed.
(ii) Monotonicity property follows from the corresponding superadditivity property (2.20), as in Theorem 2.2.
(iii) The case of concave functionf follows from the fact that the sign of the corresponding Petrović-type inequality is reversed.To conclude this section we also derive the properties of the integral Petrović-type functional, defined by (1.18).Theorem 2.5.
Supposef:I=[0,a]→ℝ is a convex function, h:[a,b]→I is continuous and monotone with h(t0)=0, where t0∈[a,b] is fixed, and let g1,g2 be functions of bounded variation with
(2.25)Gi(t):=∫atdgi(x),G¯i(t):=∫tbdgi(x)fori=1,2.
Then the functional 𝒫3, defined by (1.18), has the following properties.(i)
𝒫3(h,.;f) is subadditive with respect to functions of bounded variation, that is,
(2.26)P3(h,g1+g2;f)≤P3(h,g1;f)+P3(h,g2;f),
where ∫abh(t)dg1(t)≥0, ∫abh(t)dg2(t)≥0, and ∫abh(t)dg1(t)+∫abh(t)dg2(t)∈I.(ii)
If∫abh(t)d(g1)(t)-∫abh(t)d(g2)(t)∈I and either there exists an s≤t0 such that G1(t)≤G2(t) for t<s, G1(t)-G2(t)≥1 for s≤t≤t0, and G¯1(t)≤G¯2(t) for t>t0, or there exists an s≥t0 such that G1(t)≤G2(t) for t<t0, G¯1(t)-G¯2(t)≥1 for t0<t<s, and G¯1(t)≤G¯2(t) for t≥s, then(2.27)P3(h,g1;f)≤P3(h,g2;f).Proof.
(i) Regarding definition (1.18) of the Petrović-type integral functional, we have
(2.28)P3(h,g1+g2;f)=∫abf(h(t))d(g1+g2)(t)-f(∫abh(t)d(g1+g2)(t))-(∫abd(g1+g2)(t)-1)f(0),
that is,
(2.29)P3(h,g1+g2;f)=∫abf(h(t))dg1(t)+∫abf(h(t))dg2(t)-f(∫abh(t)dg1(t)+∫abh(t)dg2(t))-(∫abdg1(t)+∫abdg2(t)-1)f(0),
by the linearity of the differential. Now, applying inequality (1.8) to term f(∫abh(t)dg1(t)+∫abh(t)dg2(t)), we obtain
(2.30)f(∫abh(t)dg1(t)+∫abh(t)dg2(t))≥f(∫abh(t)dg1(t))+f(∫abh(t)dg2(t))-f(0).
Further, inserting (2.30) in (2.29), we have
(2.31)P3(h,g1+g2;f)≤∫abf(h(t))dg1(t)+∫abf(h(t))dg2(t)-f(∫abh(t)dg1(t))-f(∫abh(t)dg2(t))+f(0)-(∫abdg1(t)+∫abdg2(t)-1)f(0),
that is, by rearranging,
(2.32)P3(h,g1+g2;f)≤P3(h,g1;f)+P3(h,g2;f).
(ii) Monotonicity follows from the subadditivity property (2.26). Namely, representing g1 as g1=(g1-g2)+g2, we have
(2.33)P3(h,g1;f)=P3(h,(g1-g2)+g2;f)≤P3(h,g1-g2;f)+P3(h,g2;f).
Clearly, under assumptions as in the statement of theorem, we have 𝒫3(h,g1-g2;f)≤0 (see also Remark 1.11), hence it follows that 𝒫3(h,g1;f)≤𝒫3(h,g2;f), which completes the proof.


## 3.n-Exponential Convexity and Exponential Convexity of the Petrović-Type Functionals

By virtue of the results from Section2, in this section we define several new classes of Petrović-type functionals and investigate their properties regarding exponential convexity.We start these issues by giving some definitions and notions concerning exponentially convex functions which are frequently used in the results. This is a subclass of convex functions introduced by Bernstein in [2] (see also [3–5]).Definition 3.1.
A functionf:I→ℝ is n-exponentially convex in the Jensen sense on an interval I⊆ℝ, if
(3.1)∑i,j=1nξiξjf(xi+xj)≥0
holds for all choices ξi∈ℝ and xi+xj∈I,i,j=1,…,n. Function f:I→ℝ is n-exponentially convex if it is n-exponentially convex in the Jensen sense and continuous on I.The following remarks, propositions, and lemmas involvingn-exponentially convex functions are well known (see, e.g., papers [6, 7]).Remark 3.2.
It is clear from the definition that 1-exponentially convex functions in the Jensen sense are in fact nonnegative functions. Also,n-exponentially convex functions in the Jensen sense are k-exponentially convex in the Jensen sense for every k∈ℕ,k≤n.By using some linear algebra and definition of the positive semidefinite matrix, we have the following proposition.Proposition 3.3.
Iff is an n-exponentially convex in the Jensen sense then the matrix
(3.2)[f(xi+xj2)]i,j=1k
is positive semi-definite for all k∈ℕ,k≤n. In particular,
(3.3)det⁡[f(xi+xj2)]i,j=1k≥0
for all k∈ℕ,k≤n.Definition 3.4.
A functionf:I→ℝ is exponentially convex in the Jensen sense on an interval I⊆ℝ, if it is n-exponentially convex in the Jensen sense for all n∈ℕ. Moreover, function f:I→ℝ is exponentially convex if it is exponentially convex in the Jensen sense and continuous on I.Remark 3.5.
It is known (and easy to show) thatf:I→ℝ is a log-convex in the Jensen sense if and only if
(3.4)m2f(t)+2mnf(t+r2)+n2f(r)≥0
holds for each m,n∈ℝ and r,t∈I. It follows that a function is log-convex in the Jensen sense if and only if it is 2-exponentially convex in the Jensen sense. Also, using the basic convexity theory it follows that the function is log-convex if and only if it is 2-exponentially convex.We will also need the following result (see, e.g., [1]).Lemma 3.6.
IfΦ is a convex function on an interval I and if x1≤y1,x2≤y2,x1≠x2,y1≠y2, then the following inequality is valid:
(3.5)Φ(x2)-Φ(x1)x2-x1≤Φ(y2)-Φ(y1)y2-y1.
If the function Φ is concave then the sign of the above inequality is reversed.Divided differences are found to be very handy and interesting when we have to operate with different functions having different degree of smoothness. Letf:I→ℝ be a function, I an interval in ℝ. Then for distinct points ui∈I, i=0,1,2 the divided differences of first and second order are defined as follows:(3.6)[ui;f]=f(ui)(i=0,1,2),[ui,ui+1;f]=f(ui+1)-f(ui)ui+1-ui(i=0,1),[u0,u1,u2;f]=[u1,u2;f]-[u0,u1;f]u2-u0.The values of the divided differences are independent of the order of the pointsu0,u1,u2 and may be extended to include the cases when some or all points are equal, that is,(3.7)[u0,u0;f]=lim⁡u1→u0[u0,u1;f]=f′(u0),provided thatf′ exists.Now, passing through the limitu1→u0 and replacing u2 by u in (3.6), we have [1, page 16],(3.8)[u0,u0,u;f]=lim⁡u1→u0[u0,u1,u;f]=f(u)-f(u0)-f′(u0)(u-u0)(u-u0)2,u≠u0,provided thatf′ exists. Also passing to the limit ui→u(i=0,1,2) in (3.6), we have(3.9)[u,u,u;f]=lim⁡ui→u[u0,u1,u2,f]=f′′(u)2,provided thatf′′ exists.Remark 3.7.
One can note that if for allu0,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.Now, we are ready to study some new classes of Petrović-type functionals. For the sake of simplicity and to avoid many notions, we first introduce the following definitions.(M1)
Under the assumptions of Theorem1.1 equipped with conditions as in (1.1), we define linear functional as(3.10)Φ1(f)=-P1(x,p;f).(M2)
Under the assumptions of Theorem1.4 with conditions as in (1.6), we define linear functional as(3.11)Φ2(f)=Φ1(f).(M3)
Under the assumptions of Theorem1.4 with conditions as in (1.7), we define linear functional as(3.12)Φ3(f)=-Φ1(f).(M4)
Under the assumptions of Theorem2.1 with conditions as in (1.1), and provided that ∑i=1n(pi+qi)xi∈I, we define linear functional as(3.13)Φ4(f)=P1(x,p;f)+P1(x,q;f)-P1(x,p+q;f).(M5)
Under the assumptions of Theorem2.2 with conditions as in (1.6), and provided that ∑i=1n(pi+qi)xi∈I, 0<∑i=1npixi≤∑i=1nqixi, we define linear functional as(3.14)Φ5(f)=-Φ4(f).(M6)
Under the assumptions of Theorem2.2 with conditions as in (1.7), and provided that ∑i=1n(pi+qi)xi∈I, 0<∑i=1npixi≤∑i=1nqixi, we define linear functional as(3.15)Φ6(f)=Φ4(f).(M7)
Under the assumptions of Theorem1.6 with conditions as in (1.1), we define linear functional as(3.16)Φ7(f)=P2(x,p;f).(M8)
Under the assumptions of Theorem1.10 with conditions as in (1.14), we define linear functional as(3.17)Φ8(f)=P3(h,g;f).(M9)
Under the assumptions of Theorem1.10 equipped with conditions (1.16) or (1.17), we define linear functional as(3.18)Φ9(f)=-P3(h,g;f).(M10)
Under the assumptions of Theorem2.4 with conditions as in (1.1), and provided that ∑i=1n(pi+qi)xi∈I, we define linear functional as(3.19)Φ10(f)=P2(x,p+q;f)-P2(x,p;f)-P2(x,q;f).(M11)
Under the assumptions of Theorem2.5, and provided that∫abh(t)dg1(t)≥0, ∫abh(t)dg2(t)≥0, ∫abh(t)dg1(t)+∫abh(t)dg2(t)∈I, we define linear functional as
(3.20)Φ11(f)=P3(h,g1;f)+P3(h,g2;f)-P3(h,g1+g2;f).Remark 3.8.
Considering the assumptions as in(Mk), k=1,…,6, if f(u)/u is increasing function on I then
(3.21)Φk(f)≥0,fork=1,…,6.Remark 3.9.
Considering the assumptions as in(Mk), k=7,…,11, if f is convex function on I then
(3.22)Φk(f)≥0fork=7,…,11.In order to obtain our main results regarding the exponential convexity, we define different families of functions. LetI,J⊆ℝ be intervals. For distinct points u0,u1,u2∈I, we define the following.E1={ft:I→ℝ∣t∈J and t↦[u0,u1,Ft] is n-exponentially convex in the Jensen sense on J, where Ft(u)=ft(u)/u}.E2={ft:I→ℝ∣t∈J and t↦[u0,u1,u2;ft] is n-exponentially convex in the Jensen sense on J}.E3={ft:I→ℝ∣t∈J and t↦[u0,u1,Ft] is exponentially convex in the Jensen sense on J, where Ft(u)=ft(u)/u}.E4={ft:I→ℝ∣t∈J and t↦[u0,u1,u2;ft] is exponentially convex in the Jensen sense on J}.E5={ft:I→ℝ∣t∈J and t↦[u0,u1,Ft] is 2-exponentially convex in the Jensen sense on J, where Ft(u)=ft(u)/u}.E6={ft:I→ℝ∣t∈J and t↦[u0,u1,u2;ft] is 2-exponentially convex in the Jensen sense on J}.Theorem 3.10.
LetΦk(ft) be linear functionals defined as in (Mk), associated with families E1 and E2 in such a way that, ft∈E1, for k=1,…,6, and ft∈E2, for k=7,…,11. Then t↦Φk(ft) is n-exponentially convex function in the Jensen sense on J. If the function t↦Φk(ft) is continuous on J, then it is n-exponentially convex on J.Proof.
(a) We first proven-exponential convexity in the Jensen sense of the function t↦Φk(ft), for k=1,…,6. To do this, as we have considered the families of functions defined in E1, for ξi∈ℝ, i=1,…,n, and ti∈J, i=1,…,n, we define the function
(3.23)h(u)=∑i,j=1nξiξjf(ti+tj)/2(u).
Clearly, we have
(3.24)[u0,u1,H]=∑i,j=1nξiξj[u0,u1,F(ti+tj)/2],
where H(u)=h(u)/u and Ft(u)=ft(u)/u.
Since the functiont↦[u0,u1,Ft] is n-exponentially convex in the Jensen sense, the right-hand side of the above expression is nonnegative which implies that h(u)/u is increasing on I (see Remark 3.7).
Hence, taking into account Remark3.8, we have
(3.25)Φk(h)≥0,fork=1,…,6,
that is,
(3.26)∑i,j=1nξiξjΦk(f(ti+tj)/2)≥0.
Therefore, we conclude that the functions t↦Φk(ft), k=1,2,…,6, are n-exponentially convex on J in the Jensen sense.
It remains to prove then-exponential convexity in the Jensen sense of the functions t↦Φk(ft), k=7,…,11. For that sake, we consider the families of functions defined in E2. For each ξi∈ℝ, i=1,…,n, and ti∈J, i=1,…,n, we consider the function
(3.27)h(u)=∑i,j=1nξiξjf(ti+tj)/2(u).
Obviously,
(3.28)[u0,u1,u2,h]=∑i,j=1nξiξj[u0,u1,u2,f(ti+tj)/2].
Since t↦[u0,u1,u2,ft] is n-exponentially convex, the right-hand side of the above expression is nonnegative which implies that h(u) is convex on I. Moreover, taking into account Remark 3.9, we have
(3.29)Φk(h)≥0fork=7,…,11,
that is,
(3.30)∑i,j=1nξiξjΦk(f(ti+tj)/2)≥0.
Hence, t↦Φk(ft) is n-exponentially convex for k=7,…,11, and the proof is completed.The following corollary is an immediate consequence of the above theorem.Corollary 3.11.
LetΦk(ft) be linear functionals defined as in (Mk), associated with families E3 and E4 in such a way that ft∈E3, k=1,…,6, and ft∈E4, k=7,…,11. Then t↦Φk(ft) is exponentially convex function in the Jensen sense on J. If t↦Φk(ft) is continuous on J then it is exponentially convex on J.Proof.
It follows from the previous theorem.Corollary 3.12.
LetΦk(ft) be linear functionals defined as in (Mk), associated with families E5 and E6 in such a way that ft∈E5, k=1,…,6, and ft∈E6, k=7,…,11. Then the following statements hold.(i)
If the functiont↦Φk(ft) is continuous on J then it is 2-exponentially convex on J and, thus, log-convex.(ii)
If the functiont↦Φk(ft) is strictly positive and differentiable on J, then for all t,r,u,v∈J such that t≤u,r≤v, one has
(3.31)B(t,r;Φk(ft))≤B(u,v;Φk(ft)),k=1,…,11,
where(3.32)B(t,r;Φk(ft))={(Φk(ft)Φk(fr))(1/(t-r)),t≠r,exp⁡((d/dt)(Φk(ft))Φk(ft)),t=r.Proof.
(i) This is an immediate consequence of Theorem3.10 and Remark 3.2.
(ii) By (i), the functiont↦Φk(ft) is log-convex on J, which means that the function t↦logΦk(ft) is convex on J. Hence, by using Lemma 3.6 with t≤u, r≤v, t≠r, u≠v, we obtain
(3.33)log⁡⁡Φk(ft)-log⁡⁡Φk(fr)t-r≤log⁡⁡Φk(fu))-log⁡⁡Φk(fv)u-v,
that is,
(3.34)B(t,r;Φk(ft))≤B(u,v;Φk(ft)).
Finally, if t=r≤u, by taking the limit limr→t, we have
(3.35)B(t,t;Φk(ft))≤B(u,v;Φk(ft)).
Other possible cases are treated similarly.Remark 3.13.
The results given in Theorem3.10, Corollaries 3.11, and 3.12 are still valid when the points u0,u1∈I coincide, for a family of differentiable functions ft such that the function t↦[u0,u1,ft] is n-exponentially convex in the Jensen sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense). Note also that the results given in Theorem 3.10, Corollaries 3.11, and 3.12 hold when two of the points u0,u1,u2∈I coincide, say u1=u0, for a family of differentiable functions ft such that the function t↦[u0,u1,u2,ft] is n-exponentially convex in the Jensen sense (exponentially convex in the Jensen sense, log-convex in the Jensen sense). Moreover, the above results also hold when all three points coincide for a family of twice differentiable functions with the same property. These results can be proved easily by using the definition of divided differences and Remark 3.7.


## 4. Examples

We conclude this paper with several examples related to the results from the previous section.Example 4.1.
Lett>0, and let ζt:(0,∞)→ℝ be the function defined by
(4.1)ζt(u)={ut-u-log⁡⁡t,t≠1,u2,t=1.
Obviously, a family of functions ζt(u)/u is increasing for all t>0, hence, by virtue of Theorem 2.1, we obtain that the functional 𝒫1(x,.;ζt) is subadditive and decreasing on nonnegative n-tuples.Moreover, since(ζt(u)/u)′=t-u, the mapping t↦(ζt(u)/u)′ is exponentially convex (see [8]). Now, regarding Corollary 3.11 and Remark 3.13, we get exponential convexity of the functionals Φk(ζt) for k=1,…,6.In addition, Corollary3.12 provides the log-convexity of these functionals and we have(4.2)B(t,r;Φk(ζt))={(Φk(ζt)Φk(ζr))(1/(t-r)),t≠r,exp⁡(-1tlog⁡⁡t-Φk(uζt)t(Φk(ζt)),t=r≠1,exp⁡(Φk(uζ1)-2(Φk(ζ1))),t=r=1,
for k=1,2,…,6.Example 4.2.
Suppose thatt>0 and λt:(0,∞)→ℝ is the function defined by
(4.3)λt(u)=uexp⁡(-ut)-t.
Since the function λt(u)/u is increasing for every t>0, utilizing Theorem 2.1, we obtain that the functional 𝒫1(x,.;λt) is subadditive and decreasing on nonnegative n-tuples.
Further, since(λt(u)/u)′=exp(-ut), the mapping t↦(λt(u)/u)′ is exponentially convex (see [8]). Now, by using Corollary 3.11 and Remark 3.13, we get exponential convexity of the functionals Φk(λt) for k=1,…,6.
In addition, Corollary3.12 implies the log-convexity of these functionals and we have
(4.4)B(t,r;Φk(λt))={(Φk(λt)Φk(λr))(1/(t-r)),t≠r,exp⁡(-12t-Φk(uλt)2t(Φk(λt))),t=r,
for k=1,2,…,6.Example 4.3.
Consider the family of functionsψt:ℝ+→ℝ, t∈ℝ+, defined by
(4.5)ψt(u)={uexp⁡(ut)t,t≠0,u2,t=0.
It is easy to see that the function ψt(u)/u is increasing on ℝ+ for all t∈ℝ+. Hence, by virtue of Theorem 2.1, the functional 𝒫1(x,.;ψt) is subadditive and decreasing on nonnegative n-tuples.
In addition,(ψt(u)/u)′=exp(ut) and the mapping t↦(ψt(u)/u)′ is exponentially convex (see [8]). Similarly as in the previous examples, Corollary 3.11 and Remark 3.13 provide exponential convexity of the functionals Φk(ψt) for k=1,…,6.
Also, by Corollary3.12, we get log-convexity of these functionals and we have
(4.6)B(t,r;Φk(ψt))={(Φk(ψt)Φk(ψr))(1/(t-r)),t≠r,exp⁡(-1t+Φk(uψt)(Φk(ψt)),t=r≠0,exp⁡(Φk(uψ0)2(Φk(ψ0))),t=r=0,
for k=1,2,…,6.Example 4.4.
Lett>0, and let βt:(0,∞)→ℝ be the function defined by
(4.7)βt(u)={utt-1,t≠1,ulog⁡⁡u,t=1.
Obviously, a family of functions βt(u)/u is increasing for t>0, hence, by virtue of Theorem 2.1, we obtain that the functional 𝒫1(x,·;βt) is subadditive and decreasing on nonnegative n-tuples.
Further, since(βt(u)/u)′=ut-2=exp((t-2)logu), the mapping t↦(βt(u)/u)′ is exponentially convex (see [8]). Similarly as in the previous examples, regarding Corollary 3.11 and Remark 3.13, we get exponential convexity of the functionals Φk(βt) for k=1,…,6.
In addition, Corollary3.12 provides the log-convexity of these functionals and we have
(4.8)B(t,r;Φk(βt))={(Φk(βt)Φk(βr))(1/(t-r)),t≠r,exp⁡(11-t+Φk(log⁡⁡uβt)(Φk(βt)),t=r≠1,exp⁡(Φk(log⁡⁡uβ1)2(Φk(β1))),t=r=1,
for k=1,2,…,6.



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