Some Slater's Type Inequalities for Convex Functions Defined on Linear Spaces and Applications

Abstract

Some inequalities of the Slater type for convex functions defined on general linear spaces are given. Applications for norm inequalities and f-divergence measures are also provided.

1. Introduction

Suppose that 𝐼 is an interval of real numbers with interior I^∘, and 𝑓∶𝐼→ℝ is a convex function on 𝐼. Then 𝑓 is continuous on I^∘ and has finite left and right derivatives at each point of I^∘. Moreover, if 𝑥,𝑦∈I^∘ and 𝑥<𝑦, then 𝑓^′₋(𝑥)≤𝑓^′₊(𝑥)≤𝑓^′₋(𝑦)≤𝑓^′₊(𝑦) which shows that both 𝑓^′₋ and 𝑓^′₊ are nondecreasing functions on I^∘. It is also known that a convex function must be differentiable except for at most countably many points.

For a convex function 𝑓∶𝐼→ℝ, the subdifferential of 𝑓 denoted by 𝜕𝑓 is the set of all functions 𝜑∶𝐼→[−∞,∞] such that 𝜑(I^∘)⊂ℝ and𝑓(𝑥)≥𝑓(𝑎)+(𝑥−𝑎)𝜑(𝑎),forany𝑥,𝑎∈𝐼. (1.1)

It is also well known that if 𝑓 is convex on 𝐼, then 𝜕𝑓 is nonempty, 𝑓^′₋, 𝑓^′₊∈𝜕𝑓 and if 𝜑∈𝜕𝑓, then𝑓^′₋(𝑥)≤𝜑(𝑥)≤𝑓^′₊(𝑥),forany𝑥∈I^∘. (1.2) In particular, 𝜑 is a nondecreasing function.

If 𝑓 is differentiable and convex on I^∘, then 𝜕𝑓={𝑓^′}.

The following result is well known in the literature as the Slater inequality.

Theorem 1.1 (Slater, 1981, [1]). If 𝑓∶𝐼→ℝ is a nonincreasing (nondecreasing) convex function, 𝑥_𝑖∈𝐼,𝑝_𝑖≥0 with 𝑃_𝑛∶=∑^𝑛_𝑖=1𝑝_𝑖>0 and ∑^𝑛_𝑖=1𝑝_𝑖𝜑(𝑥_𝑖)≠0, where 𝜑∈𝜕𝑓, then 1𝑃_𝑛^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖∑≤𝑓^𝑛𝑖=1𝑝_𝑖𝑥_𝑖𝜑𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝜑𝑥_𝑖. (1.3)

As pointed out in [2] (see also [3, p. 64] and [4, p. 208]), the monotonicity assumption for the derivative 𝜑 can be replaced with the condition ∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖𝜑𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝜑𝑥_𝑖∈𝐼, (1.4) which is more general and can hold for suitable points in 𝐼 and for not necessarily monotonic functions.

For recent works on Slater’s inequality, see [5–7].

The main aim of the present paper is to extend Slater’s inequality for convex functions defined on general linear spaces. A reverse of the Slater’s inequality is also obtained. Natural applications for norm inequalities and 𝑓-divergence measures are provided as well.

2. Slater’s Inequality for Functions Defined on Linear Spaces

Assume that 𝑓∶𝑋→ℝ is a convex function on the real linear space 𝑋. Since for any vectors 𝑥,𝑦∈𝑋 the function 𝑔_𝑥,𝑦∶ℝ→ℝ,𝑔_𝑥,𝑦(𝑡)∶=𝑓(𝑥+𝑡𝑦) is convex, it follows that the following limits exist∇₊₍₋₎𝑓(𝑥)𝑦∶=lim_{(−)𝑡→0+}𝑓𝑥+𝑡𝑦−𝑓(𝑥)𝑡, (2.1) and they are called the right (left) Gâteaux derivatives of the function 𝑓 in the point 𝑥 over the direction 𝑦.

It is obvious that for any 𝑡>0>𝑠 we have𝑓𝑥+𝑡𝑦−𝑓(𝑥)𝑡≥∇₊𝑓(𝑥)𝑦=inf_𝑡>0𝑓𝑥+𝑡𝑦−𝑓(𝑥)𝑡≥sup_𝑠<0𝑓𝑥+𝑠𝑦−𝑓(𝑥)𝑠=∇₋𝑓(𝑥)𝑦≥𝑓𝑥+𝑠𝑦−𝑓(𝑥)𝑠, (2.2) for any 𝑥,𝑦∈𝑋 and, in particular,∇₋𝑓(𝑢)(𝑢−𝑣)≥𝑓(𝑢)−𝑓(𝑣)≥∇₊𝑓(𝑣)(𝑢−𝑣), (2.3) for any 𝑢,𝑣∈𝑋. We call this the gradient inequality for the convex function 𝑓. It will be used frequently in the sequel in order to obtain various results related to Slater’s inequality.

The following properties are also of importance:∇₊𝑓(𝑥)−𝑦=−∇₋𝑓(𝑥)𝑦,∇₊₍₋₎𝑓(𝑥)𝛼𝑦=𝛼∇₊₍₋₎𝑓(𝑥)𝑦, (2.4)(2.5) for any 𝑥,𝑦∈𝑋 and 𝛼≥0.

The right Gâteaux derivative is subadditive while the left one is superadditive, that is,∇₊𝑓(𝑥)𝑦+𝑧≤∇₊𝑓(𝑥)𝑦+∇₊𝑓(𝑥𝑧),∇)(₋𝑓(𝑥)𝑦+𝑧≥∇₋𝑓(𝑥)𝑦+∇₋𝑓(𝑥)(𝑧), (2.6) for any 𝑥,𝑦,𝑧∈𝑋.

Some natural examples can be provided by the use of normed spaces.

Assume that (𝑋,‖⋅‖) is a real normed linear space. The function 𝑓∶𝑋→ℝ, 𝑓(𝑥)∶=(1/2)‖𝑥‖² is a convex function which generates the superior and the inferior semi-inner products 𝑦,𝑥_𝑠(𝑖)∶=lim_{(−)𝑡→0+}𝑥+𝑡𝑦²−‖𝑥‖²2𝑡. (2.7) For a comprehensive study of the properties of these mappings in the Geometry of Banach Spaces, see the monograph [8].

For the convex function 𝑓_𝑝∶𝑋→ℝ, 𝑓_𝑝(𝑥)∶=‖𝑥‖^𝑝 with 𝑝>1, we have∇₊₍₋₎𝑓_𝑝(𝑥)𝑦=𝑝‖𝑥‖^𝑝−2𝑦,𝑥_𝑠(𝑖)if𝑥≠0,0if𝑥=0, (2.8) for any 𝑦∈𝑋.

If 𝑝=1, then we have∇₊₍₋₎𝑓₁(𝑥)𝑦=‖𝑥‖⁻¹𝑦,𝑥_𝑠(𝑖)+(−𝑦if𝑥≠0,)if𝑥=0, (2.9) for any 𝑦∈𝑋.

For a given convex function 𝑓∶𝑋→ℝ and a given 𝑛-tuple of vectors 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛, we consider the setsSla₊₍₋₎𝑓,𝐱∶=𝑣∈𝑋∣∇₊₍₋₎𝑓𝑥_𝑖𝑣−𝑥_𝑖≥0∀𝑖∈{1,…,𝑛},Sla₊₍₋₎𝑓,𝐱,𝐩∶=𝑣∈𝑋∣^𝑛_𝑖=1𝑝_𝑖∇₊₍₋₎𝑓𝑥_𝑖𝑣−𝑥_𝑖≥0, (2.10) where 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 is a given probability distribution, that is, 𝑝_𝑖≥0 for 𝑖∈{1,…,𝑛} and ∑^𝑛_𝑖=1𝑝_𝑖=1.

The following properties of these sets hold.

Lemma 2.1. For a given convex function 𝑓∶𝑋→ℝ, a given 𝑛-tuple of vectors 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛, and a given probability distribution 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛, one has (i)Sla₋(𝑓,𝐱)⊂Sla₊(𝑓,𝐱) and Sla₋(𝑓,𝐱,𝐩)⊂Sla₊(𝑓,𝐱,𝐩);(ii)Sla₋(𝑓,𝐱)⊂Sla₋(𝑓,𝐱,𝐩) and Sla₊(𝑓,𝐱)⊂Sla(𝑓,𝐱,𝐩) for all 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛;(iii) the sets Sla₋(𝑓,𝐱) and Sla₋(𝑓,𝐱,𝐩) are convex.

Proof. The properties (i) and (ii) follow from the definition and the fact that ∇₊𝑓(𝑥)(𝑦)≥∇₋𝑓(𝑥)(𝑦) for any 𝑥,𝑦.
(iii) Let us only prove that Sla₋(𝑓,𝐱) is convex.
If we assume that 𝑦₁,𝑦₂∈Sla₋(𝑓,𝐱) and 𝛼,𝛽∈[0,1] with 𝛼+𝛽=1, then by the superadditivity and positive homogeneity of the Gâteaux derivative ∇₋𝑓(⋅)(⋅) in the second variable we have ∇₋𝑓𝑥_𝑖𝛼𝑦₁+𝛽𝑦₂−𝑥_𝑖=∇₋𝑓𝑥_𝑖𝛼𝑦₁−𝑥_𝑖+𝛽𝑦₂−𝑥_𝑖≥𝛼∇₋𝑓𝑥_𝑖𝑦₁−𝑥_𝑖+𝛽∇₋𝑓𝑥_𝑖𝑦₂−𝑥_𝑖≥0, (2.11) for all 𝑖∈{1,…,𝑛}, which shows that 𝛼𝑦₁+𝛽𝑦₂∈Sla₋(𝑓,𝐱)
The proof for the convexity of Sla₋(𝑓,𝐱,𝐩) is similar and the details are omitted.

For the convex function 𝑓_𝑝∶𝑋→ℝ, 𝑓_𝑝(𝑥)∶=‖𝑥‖^𝑝 with 𝑝≥1, defined on the normed linear space (𝑋,‖⋅‖) and for the 𝑛-tuple of vectors 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛⧵{(0,…,0)} we have, by the well-known property of the semi-inner products,𝑦+𝛼𝑥,𝑥_𝑠(𝑖)=𝑦,𝑥_𝑠(𝑖)+𝛼‖𝑥‖²,forany𝑥,𝑦∈𝑋,𝛼∈ℝ, (2.12) thatSla₊₍₋₎‖⋅‖^𝑝,𝐱=Sla₊₍₋₎(‖⋅‖,𝐱)∶=𝑣∈𝑋∣𝑣,𝑥_{𝑗𝑠(𝑖)}≥𝑥_𝑗²∀𝑗∈{1,…,𝑛} (2.13) which, as can be seen, does not depend on 𝑝. We observe, by the continuity of the semi-inner products in the first variable, that Sla₊₍₋₎(‖⋅‖,𝐱) is closed in (𝑋,‖⋅‖). Also, we should remark that if 𝑣∈Sla₊₍₋₎(‖⋅‖,𝐱), then for any 𝛾≥1 we also have that 𝛾𝑣∈Sla₊₍₋₎(‖⋅‖,𝐱).

The larger classes, which are dependent on the probability distribution 𝐩∈ℙ^𝑛, are described bySla₊₍₋₎‖⋅‖^𝑝,𝐱,𝐩∶=𝑣∈𝑋∣^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_{𝑗𝑠(𝑖)}≥^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝. (2.14) If the normed space is smooth, that is, the norm is Gâteaux differentiable in any nonzero point, then the superior and inferior semi-inner products coincide with the Lumer-Giles semi-inner product [⋅,⋅] that generates the norm and is linear in the first variable (see for instance [8]). In this situation,Sla(‖⋅‖,𝐱)=𝑣∈𝑋∣𝑣,𝑥_𝑗≥𝑥_𝑗²{,‖⋅‖∀𝑗∈1,…,𝑛}Sla^𝑝,𝐱,𝐩=𝑣∈𝑋∣^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗≥^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝. (2.15) If (𝑋,⟨⋅,⋅⟩) is an inner product space, then Sla(‖⋅‖^𝑝,𝐱,𝐩) can be described bySla‖⋅‖^𝑝,𝐱,𝐩=𝑣∈𝑋∣𝑣,^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑥_𝑗≥^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝, (2.16) and if the family {𝑥_𝑗}_{𝑗=1,…,𝑛} is orthogonal, then obviously, by the Pythagoras theorem, we have that the sum ∑^𝑛_𝑗=1𝑥_𝑗 belongs to Sla(‖⋅‖,𝐱) and therefore to Sla(‖⋅‖^𝑝,𝐱,𝐩) for any 𝑝≥1 and any probability distribution 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛.

We can state now the following results that provide a generalization of Slater’s inequality as well as a counterpart for it.

Theorem 2.2. Let 𝑓∶𝑋→ℝ be a convex function on the real linear space 𝑋, 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛 an 𝑛-tuple of vectors, and 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 a probability distribution. Then for any 𝑣∈Sla₊(𝑓,𝐱,𝐩), one has the inequalities ∇₋𝑓(𝑣)(𝑣)−^𝑛_𝑖=1𝑝_𝑖∇₋𝑓(𝑣)𝑥_𝑖≥𝑓(𝑣)−^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖≥0. (2.17)

Proof. If we write the gradient inequality for 𝑣∈Sla₊(𝑓,𝐱,𝐩) and 𝑥_𝑖, then we have that ∇₋𝑓(𝑣)𝑣−𝑥_𝑖≥𝑓(𝑣)−𝑓𝑥_𝑖≥∇₊𝑓𝑥_𝑖𝑣−𝑥_𝑖, (2.18) for any 𝑖∈{1,…,𝑛}.
By multiplying (2.18) with 𝑝_𝑖≥0 and summing over 𝑖 from 1 to 𝑛, we get ^𝑛_𝑖=1𝑝_𝑖∇₋𝑓(𝑣)𝑣−𝑥_𝑖≥𝑓(𝑣)−^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖≥^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑣−𝑥_𝑖. (2.19) Now, since 𝑣∈Sla₊(𝑓,𝐱,𝐩), then the right hand side of (2.19) is nonnegative, which proves the second inequality in (2.17).
By the superadditivity of the Gâteaux derivative ∇₋𝑓(⋅)(⋅) in the second variable, we have ∇₋𝑓(𝑣)(𝑣)−∇₋𝑓(𝑣)𝑥_𝑖≥∇₋𝑓(𝑣)𝑣−𝑥_𝑖, (2.20) which, by multiplying with 𝑝_𝑖≥0 and summing over 𝑖 from 1 to 𝑛, produces the inequality ∇₋𝑓(𝑣)(𝑣)−^𝑛_𝑖=1𝑝_𝑖∇₋𝑓(𝑣)𝑥_𝑖≥^𝑛_𝑖=1𝑝_𝑖∇₋𝑓(𝑣)𝑣−𝑥_𝑖. (2.21) Utilising (2.19) and (2.21), we deduce the desired result (2.17).

Remark 2.3. The above result has the following form for normed linear spaces. Let (𝑋,‖⋅‖) be a normed linear space, 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛 an 𝑛-tuple of vectors from 𝑋, and 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 a probability distribution. Then for any vector 𝑣∈𝑋 with the property ^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗𝑠≥^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝,𝑝≥1, (2.22) we have the inequalities 𝑝‖𝑣‖^𝑝−^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗𝑖≥‖𝑣‖^𝑝−^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝≥0. (2.23)
Rearranging the first inequality in (2.23), we also have that 𝑝−1‖𝑣‖^𝑝+^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝≥𝑝^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗𝑖. (2.24)
If the space is smooth, then the condition (2.22) becomes ^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗≥^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝,𝑝≥1, (2.25) implying the inequality 𝑝‖𝑣‖^𝑝−^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗≥‖𝑣‖^𝑝−^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝≥0. (2.26) Notice also that the first inequality in (2.26) is equivalent with 𝑝−1‖𝑣‖^𝑝+^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝≥𝑝^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−2𝑣,𝑥_𝑗≥𝑝^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝≥0. (2.27)

Corollary 2.4. Let 𝑓∶𝑋→ℝ be a convex function on the real linear space 𝑋, 𝑥=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛 an 𝑛-tuple of vectors, and 𝑝=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 a probability distribution. If ^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖≥(<)0, (2.28) and there exists a vector 𝑠∈𝑋 with ^𝑛_𝑖=1𝑝_𝑖∇₊₍₋₎𝑓𝑥_𝑖(𝑠)≥(≤)1, (2.29) then ∇₋𝑓^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠−^𝑛_𝑖=1𝑝_𝑖∇₋𝑓^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠𝑥_𝑖≥𝑓^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠−^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖≥0. (2.30)

Proof. Assume that ∑^𝑛_𝑖=1𝑝_𝑖∇₊𝑓(𝑥_𝑖)(𝑥_𝑖)≥0 and ∑^𝑛_𝑖=1𝑝_𝑖∇₊𝑓(𝑥_𝑖)(𝑠)≥1 and define 𝑣∶=∑^𝑛_𝑗=1𝑝_𝑗∇₊𝑓(𝑥_𝑗)(𝑥_𝑗)𝑠. We claim that 𝑣∈Sla₊(𝑓,𝐱,𝐩).
By the subadditivity and positive homogeneity of the mapping ∇₊𝑓(⋅)(⋅) in the second variable, we have ^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑣−𝑥_𝑖≥^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖(𝑣)−^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖=^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠−^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖=^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖(𝑠)−^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖=^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖(𝑠)−1≥0, (2.31) as claimed. Applying Theorem 2.2 for this 𝑣, we get the desired result.
If ∑^𝑛_𝑖=1𝑝_𝑖∇₊𝑓(𝑥_𝑖)(𝑥_𝑖)<0 and ∑^𝑛_𝑖=1𝑝_𝑖∇₋𝑓(𝑥_𝑖)(𝑠)≤1, then for 𝑤∶=^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠, (2.32) we also have that ^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑤−𝑥_𝑖≥^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗𝑠−^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖=^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖−^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗(−𝑠)−^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖=−^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖(−𝑠)−^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖𝑥_𝑖=−^𝑛_𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗1+^𝑛_𝑖=1𝑝_𝑖∇₊𝑓𝑥_𝑖()=−−𝑠_𝑛𝑗=1𝑝_𝑗∇₊𝑓𝑥_𝑗𝑥_𝑗1−^𝑛_𝑖=1𝑝_𝑖∇₋𝑓𝑥_𝑖(𝑠)≥0, (2.33) where, for the last equality, we have used the property (2.4). Therefore, 𝑤∈Sla₊(𝑓,𝐱,𝐩) and by Theorem 2.2 we get the desired result.

It is natural to consider the case of normed spaces.

Remark 2.5. Let (X,‖⋅‖) be a normed linear space, 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝑋^𝑛 an n-tuple of vectors from 𝑋, and 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 a probability distribution. Then for any vector 𝑠∈𝑋 with the property that 𝑝^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖^𝑝−2𝑠,𝑥_𝑖𝑠≥1, (2.34) we have the inequalities 𝑝^𝑝‖𝑠‖^𝑝−1𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝𝑝−1𝑝‖𝑠‖^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝−^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗,𝑠_𝑖≥𝑝^𝑝‖𝑠‖^𝑝𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝𝑝−^𝑛_𝑗=1𝑝_𝑗𝑥_𝑗^𝑝≥0. (2.35)
The case of smooth spaces can be easily derived from the above; however, the details are left to the interested reader.

3. The Case of Finite Dimensional Linear Spaces

Consider now the finite dimensional linear space 𝑋=ℝ^𝑚 and assume that 𝐶 is an open convex subset of ℝ^𝑚. Assume also that the function 𝑓∶𝐶→ℝ is differentiable and convex on 𝐶. Obviously, if 𝑥=(𝑥¹,…,𝑥^𝑚)∈𝐶, then for any 𝑦=(𝑦¹,…,𝑦^𝑚)∈ℝ^𝑚 we have∇𝑓(𝑥)𝑦=^𝑚_𝑘=1𝜕𝑓𝑥¹,…,𝑥^𝑚𝜕𝑥^𝑘⋅𝑦^𝑘. (3.1) For the convex function 𝑓∶𝐶→ℝ and a given 𝑛-tuple of vectors 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝐶^𝑛 with 𝑥_𝑖=(𝑥¹_𝑖,…,𝑥^𝑚_𝑖) with 𝑖∈{1,…,𝑛}, we consider the setsSla𝑓,𝐱,𝐶∶=𝑣∈𝐶∣^𝑚_𝑘=1𝜕𝑓𝑥¹_𝑖,…,𝑥^𝑚_𝑖𝜕𝑥^𝑘_𝑖⋅𝑣^𝑘≥^𝑚_𝑘=1𝜕𝑓𝑥¹_𝑖,…,𝑥^𝑚_𝑖𝜕𝑥^𝑘_𝑖⋅𝑥^𝑘_𝑖∀𝑖∈{1,…,𝑛},Sla𝑓,𝐱,𝐩,𝐶∶=𝑣∈𝐶∣^𝑛𝑚_𝑖=1𝑘=1𝑝_𝑖𝜕𝑓𝑥¹_𝑖,…,𝑥^𝑚_𝑖𝜕𝑥^𝑘_𝑖⋅𝑣^𝑘≥^𝑛𝑚_𝑖=1𝑘=1𝑝_𝑖𝜕𝑓𝑥¹_𝑖,…,𝑥^𝑚_𝑖𝜕𝑥^𝑘_𝑖⋅𝑥^𝑘_𝑖, (3.2) where 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 is a given probability distribution.

As in the previous section the sets, Sla(𝑓,𝐱,𝐶) and Sla(𝑓,𝐱,𝐩,𝐶) are convex and closed subsets of clo(𝐶), the closure of 𝐶. Also {𝑥₁,…,𝑥_𝑛}⊂Sla(𝑓,𝐱,𝐶)⊂Sla(𝑓,𝐱,𝐩,𝐶) for any 𝐩=(𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛is a probability distribution.

Proposition 3.1. Let 𝑓∶𝐶→ℝ be a convex function on the open convex set 𝐶 in the finite dimensional linear space ℝ^𝑚, (𝑥₁,…,𝑥_𝑛)∈𝐶^𝑛 an 𝑛-tuple of vectors and (𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 a probability distribution. Then for any 𝑣=(𝑣¹,…,𝑣^𝑚)∈Sla(𝑓,𝐱,𝐩,𝐶), one has the inequalities 𝜕𝑓𝑣^𝑚_𝑘=1¹,…,𝑣^𝑚𝜕𝑥^𝑘⋅𝑣^𝑘−^𝑛𝑚_𝑖=1𝑘=1𝑝_𝑖𝜕𝑓𝑥¹_𝑖,…,𝑥^𝑚_𝑖𝜕𝑥^𝑘_𝑖⋅𝑣^𝑘≥𝑓𝑣¹,…,𝑣^𝑛−^𝑛_𝑖=1𝑝_𝑖𝑓𝑥¹_𝑖,…,𝑥^𝑚_𝑖≥0. (3.3)

The unidimensional case, that is, 𝑚=1 is of interest for applications. We will state this case with the general assumption that 𝑓∶𝐼→ℝ is a convex function on an open interval 𝐼. For a given 𝑛-tuple of vectors 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝐼^𝑛, we have Sla₊₍₋₎𝑓,𝐱,𝐼∶=𝑣∈𝐼∣𝑓^′₊₍₋₎𝑥_𝑖⋅𝑣−𝑥_𝑖≥0∀𝑖∈{1,…,𝑛},Sla₊₍₋₎𝑓,𝐱,𝐩,𝐈∶=𝑣∈𝐼∣^𝑛_𝑖=1𝑝_𝑖𝑓^′₊₍₋₎𝑥_𝑖⋅𝑣−𝑥_𝑖≥0, (3.4) where (𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛  is a probability distribution. These sets inherit the general properties pointed out in Lemma 2.1. Moreover, if we make the assumption that ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊(𝑥_𝑖)≠0, then for ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊(𝑥_𝑖)>0 we have Sla₊=∑𝑓,𝐱,𝐩,𝐈𝑣∈𝐼∣𝑣≥^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖, (3.5) while for ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊(𝑥_𝑖)<0 we have Sla₊=∑𝑓,𝐱,𝐩,𝐈𝑣∈𝐼∣𝑣≤^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖. (3.6)

Also, if we assume that 𝑓^′₊(𝑥_𝑖)≥0 for all 𝑖∈{1,…,𝑛} and ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊(𝑥_𝑖)>0, then 𝑣_𝑠∑∶=^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖∈𝐼, (3.7) due to the fact that 𝑥_𝑖∈𝐼 and 𝐼 is a convex set.

Proposition 3.2. Let 𝑓∶𝐼→ℝ be a convex function on an open interval 𝐼. For a given 𝑛-tuple of vectors 𝐱=(𝑥₁,…,𝑥_𝑛)∈𝐼^𝑛 and (𝑝₁,…,𝑝_𝑛)∈ℙ^𝑛 a probability distribution, one has 𝑓^′₋(𝑣)𝑣−^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖≥𝑓(𝑣)−^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖≥0, (3.8) for any 𝑣∈Sla₊(𝑓,𝐱,𝐩,𝐈).
In particular, if one assumes that ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊(𝑥_𝑖)≠0 and ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖∈𝐼, (3.9) then 𝑓^′₋∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖−∑^𝑛_𝑖=1𝑝_𝑖𝑥_𝑖∑≥𝑓^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖𝑥_𝑖∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑥_𝑖−^𝑛_𝑖=1𝑝_𝑖𝑓𝑥_𝑖≥0. (3.10)
Moreover, if 𝑓^′₊(𝑥_𝑖)≥0 for all 𝑖∈{1,…,𝑛} and ∑^𝑛_𝑖=1𝑝_𝑖𝑓^′₊(𝑥_𝑖)>0, then (3.10) holds true as well.

Remark 3.3. We remark that the first inequality in (3.10) provides a reverse inequality for the classical result due to Slater.

4. Some Applications for 𝑓-Divergences

Given a convex function 𝑓∶[0,∞)→ℝ, the 𝑓-divergence functional 𝐼_𝑓(𝐩,𝐪)∶=^𝑛_𝑖=1𝑞_𝑖𝑓𝑝_𝑖𝑞_𝑖, (4.1) where 𝐩=(𝑝₁,…,𝑝_𝑛),𝐪=(𝑞₁,…,𝑞_𝑛) are positive sequences, was introduced by Csiszár in [9], as a generalized measure of information, a “distance function’’ on the set of probability distributions ℙ^𝑛. As in [9], we interpret undefined expressions by𝑓(0)=lim_𝑡→0+𝑓(𝑡)0,0𝑓0𝑎=0,0𝑓0=lim_𝑞→0+𝑎𝑞𝑓𝑞=𝑎lim_𝑡→∞𝑓(𝑡)𝑡,𝑎>0. (4.2) The following results were essentially given by Csiszár and Körner [10]:(i)if 𝑓 is convex, then 𝐼_𝑓(𝐩,𝐪) is jointly convex in 𝐩 and 𝐪; (ii)for every 𝐩,𝐪∈ℝ^𝑛₊, we have𝐼_𝑓(𝐩,𝐪)≥^𝑛_𝑗=1𝑞_𝑗𝑓∑^𝑛_𝑗=1𝑝_𝑗∑^𝑛_𝑗=1𝑞_𝑗. (4.3)

If 𝑓 is strictly convex, equality holds in (4.3) if and only if𝑝₁𝑞₁=𝑝₂𝑞₂𝑝=⋯=^𝑛𝑞_𝑛. (4.4)

If 𝑓 is normalized, that is, 𝑓(1)=0, then for every 𝐩,𝐪∈ℝ^𝑛₊ with ∑^𝑛_𝑖=1𝑝_𝑖=∑^𝑛_𝑖=1𝑞_𝑖, we have the inequality𝐼_𝑓(𝐩,𝐪)≥0. (4.5)

In particular, if 𝐩,𝐪∈ℙ^𝑛, then (4.5) holds. This is the well-known positivity property of the 𝑓-divergence.

It is obvious that the above definition of 𝐼_𝑓(𝐩,𝐪) can be extended to any function 𝑓∶[0,∞)→ℝ; however, the positivity condition will not generally hold for normalized functions and 𝐩,𝐪∈ℝ^𝑛₊ with ∑^𝑛_𝑖=1𝑝_𝑖=∑^𝑛_𝑖=1𝑞_𝑖.

For a normalized convex function 𝑓∶[0,∞)→ℝ and two probability distributions 𝐩,𝐪∈ℙ^𝑛, we define the setSla₊𝑓,𝐩,𝐪∶=𝑣∈[0,∞)∣^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖𝑞_𝑖⋅𝑝𝑣−^𝑖𝑞_𝑖≥0. (4.6) Now, observe that^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖𝑞_𝑖⋅𝑝𝑣−^𝑖𝑞_𝑖≥0, (4.7) is equivalent with𝑣^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖𝑞_𝑖≥^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖𝑞_𝑖. (4.8) If ∑^𝑛_𝑖=1𝑞_𝑖𝑓^′₊(𝑝_𝑖/𝑞_𝑖)>0, then (4.8) is equivalent with∑𝑣≥^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖∑^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖, (4.9) therefore in this caseSla₊=[)𝑓,𝐩,𝐪0,∞if^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖∑<0,_𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖∑^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖,∞if^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖≥0. (4.10)

If ∑^𝑛_𝑖=1𝑞_𝑖𝑓^′₊(𝑝_𝑖/𝑞_𝑖)<0, then (4.8) is equivalent with∑𝑣≤^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖∑^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖, (4.11) thereforeSla₊=∑𝑓,𝐩,𝐪0,^𝑛𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖∑^𝑛_𝑖=1𝑞_𝑖𝑓^′₊𝑝_𝑖/𝑞_𝑖if^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖𝑞_𝑖≤0,∅if^𝑛_𝑖=1𝑝_𝑖𝑓^′₊𝑝_𝑖𝑞_𝑖>0. (4.12)

Utilising the extended 𝑓-divergences notation, we can state the following result.

Theorem 4.1. Let 𝑓∶[0,∞)→ℝ be a normalized convex function and 𝐩,𝐪∈ℙ^𝑛 two probability distributions. If 𝑣∈Sla₊(𝑓,𝐩,𝐪), then one has 𝑓^′₋(𝑣)(𝑣−1)≥𝑓(𝑣)−𝐼_𝑓(𝐩,𝐪)≥0. (4.13) In particular, if one assumes that 𝐼_𝑓′+(𝐩,𝐪)≠0 and 𝐼_{𝑓′+(⋅)(⋅)}(𝐩,𝐪)𝐼_𝑓′+()∈[)𝐩,𝐪0,∞, (4.14) then 𝑓^′₋𝐼_{𝑓′+(⋅)(⋅)}(𝐩,𝐪)𝐼_𝑓′+()𝐼𝐩,𝐪^{𝑓′+(⋅)(⋅)}(𝐩,𝐪)𝐼_𝑓′+()𝐼𝐩,𝐪−1≥𝑓^{𝑓′+(⋅)(⋅)}(𝐩,𝐪)𝐼_𝑓′+(𝐩,𝐪)−𝐼_𝑓(𝐩,𝐪)≥0. (4.15) Moreover, if 𝑓^′₊(𝑝_𝑖/𝑞_𝑖)≥0 for all 𝑖∈{1,…,𝑛} and 𝐼_𝑓′+(𝐩,𝐪)>0, then (4.15) holds true as well.

The proof follows immediately from Proposition 3.2 and the details are omitted.

The K. Pearson 𝜒²-divergence is obtained for the convex function 𝑓(𝑡)=(1−𝑡)²,𝑡∈ℝ and given by 𝜒²(𝐩,𝐪)∶=^𝑛_𝑗=1𝑞_𝑗𝑝_𝑗𝑞_𝑗−1²=^𝑛_𝑗=1𝑝_𝑗−𝑞_𝑗²𝑞_𝑗=^𝑛_𝑗=1𝑝²_𝑖𝑞_𝑖−1. (4.16)

The Kullback-Leibler divergence can be obtained for the convex function 𝑓∶(0,∞)→ℝ, 𝑓(𝑡)=𝑡ln𝑡 and is defined by KL(𝐩,𝐪)∶=^𝑛_𝑗=1𝑞_𝑗⋅𝑝_𝑗𝑞_𝑗𝑝ln^𝑗𝑞_𝑗=^𝑛_𝑗=1𝑝_𝑗𝑝ln^𝑗𝑞_𝑗. (4.17)

If we consider the convex function 𝑓∶(0,∞)→ℝ, 𝑓(𝑡)=−ln𝑡, then we observe that 𝐼_𝑓(𝐩,𝐪)∶=^𝑛_𝑖=1𝑞_𝑖𝑓𝑝_𝑖𝑞_𝑖=−^𝑛_𝑖=1𝑞_𝑖𝑝ln^𝑖𝑞_𝑖=^𝑛_𝑖=1𝑞_𝑖𝑞ln^𝑖𝑝_𝑖=KL(𝐪,𝐩). (4.18)

For the function 𝑓(𝑡)=−ln𝑡, we will obviously have that Sla(−ln,𝐩,𝐪)∶=𝑣∈[0,∞)∣−^𝑛_𝑖=1𝑞_𝑖𝑝_𝑖𝑞_𝑖⁻¹⋅𝑝𝑣−^𝑖𝑞_𝑖=[)≥0𝑣∈0,∞∣𝑣^𝑛_𝑖=1𝑞²_𝑖𝑝_𝑖=1−1≤00,𝜒²().𝐪,𝐩+1 (4.19) Utilising the first part of Theorem 4.1, we can state the following.

Proposition 4.2. Let 𝐩,𝐪∈ℙ^𝑛 be two probability distributions. If 𝑣∈[0,(1/(𝜒²(𝐪,𝐩)+1))], then one has 1−𝑣𝑣≥−ln(𝑣)−KL(𝐪,𝐩)≥0. (4.20) In particular, for 𝑣=1/(𝜒²(𝐪,𝐩)+1), one gets 𝜒²(𝐪,𝐩)≥ln𝜒²(𝐪,𝐩)+1−KL(𝐪,𝐩)≥0. (4.21)

If we consider now the function 𝑓∶(0,∞)→ℝ, 𝑓(𝑡)=𝑡ln𝑡, then 𝑓^′(𝑡)=ln𝑡+1 and Sla((⋅)ln(⋅),𝐩,𝐪)∶=𝑣∈[0,∞)∣^𝑛_𝑖=1𝑞_𝑖𝑝ln^𝑖𝑞_𝑖⋅𝑝+1𝑣−^𝑖𝑞_𝑖=[)≥0𝑣∈0,∞∣𝑣^𝑛_𝑖=1𝑞_𝑖𝑝ln^𝑖𝑞_𝑖+1−^𝑛_𝑖=1𝑝_𝑖⋅𝑝ln^𝑖𝑞_𝑖={[)(((.+1≥0𝑣∈0,∞∣𝑣1−KL𝐪,𝐩))≥1+KL𝐩,𝐪)} (4.22) We observe that if 𝐩,𝐪∈ℙ^𝑛 are two probability distributions such that 0<KL(𝐪,𝐩)<1, then ⋅)(⋅))=()Sla((ln,𝐩,𝐪1+KL𝐩,𝐪1−KL(𝐪,𝐩),∞. (4.23) If KL(𝐪,𝐩)≥1, then Sla((⋅)ln(⋅),𝐩,𝐪)=∅.

By the use of Theorem 4.1, we can state now the following.

Proposition 4.3. Let 𝐩,𝐪∈ℙ^𝑛 be two probability distributions such that 0<KL(𝐪,𝐩)<1. If 𝑣∈[(1+KL(𝐩,𝐪))/(1−KL(𝐪,𝐩)),∞), then one has (ln𝑣+1)(𝑣−1)≥𝑣ln(𝑣)−KL(𝐩,𝐪)≥0. (4.24) In particular, for 𝑣=(1+KL(𝐩,𝐪))/(1−KL(𝐪,𝐩)), one gets ()ln1+KL𝐩,𝐪()()1−KL𝐪,𝐩+11+KL𝐩,𝐪()≥()1−KL𝐪,𝐩−11+KL𝐩,𝐪()()1−KL𝐪,𝐩ln1+KL𝐩,𝐪()()1−KL𝐪,𝐩−KL𝐩,𝐪≥0. (4.25)

Similar results can be obtained for other divergence measures of interest such as the Jeffreys divergence and Hellinger discrimination. However, the details are left to the interested reader.

Acknowledgment

The author would like to thank the anonymous referees for their valuable comments that have been implemented in the final version of the paper.