Jensen's Inequality for Quasiconvex Functions

Some inequalities of Jensen type and connected results 
are given for quasiconvex functions on convex sets in real linear spaces.


Introduction
Throughout this paper X denotes a real linear space and C ⊆ X a convex set, so that x, y ∈ C with λ ∈ [0, 1] implies that λx + (1 − λ)y ∈ C. This class of functions strictly contains the class of convex functions defined on a convex set in a real linear space. See [8] and citations therein for an overview of this issue.
Some recent studies have shown that quasiconvex functions have quite close resemblances to convex functions -see, for example, [4], [6], [7], [10] for quasiconvex and even more general extensions of convex functions in the context of Hadamard's pair of inequalities. Apart from generalizations to theory, weakening the convexity condition can increase applicability. Thus in [9] use is made of quasiconvexity to obtain a global extremum with rather less effort than via convexity. In this article we pursue the concept further and derive a number of Jensen-type inequalities for quasiconvex functions. See also [5] for functions of Godunova-Levin type in the context of Jensen's inequality.

Preliminaries
For an arbitrary mapping f : C → IR and x, y two fixed elements in C, we can define the map g x,y : [0, 1] → IR by g x,y (t) = f (tx + (1 − t)y). This provides a characterization of quasiconvexity.
Proposition 2.1 The following statements are equivalent: (i) f is quasiconvex on C; (ii) for every x, y ∈ C, the mapping g x,y is quasiconvex on [0, 1].
For the reverse implication, suppose (ii) holds. Then which shows that f is quasiconvex on C.
Proof. From the given conditions, for each t ∈ [0, 1], For a given mapping f : C → IR we may also define a map G t : Again we have a characterization of quasiconvexity.

Proposition 2.4
We have the following: which shows that G t is quasiconvex on C 2 .
Proof. We employ induction on n. The case n = 1 provides a trivial basis. Assume that the stated inequality holds for n = 1, . . . , k (k ≥ 1). By quasiconvexity and the inductive assumption This may be written as the result of the theorem with n = k+1, giving the inductive step and so completing the proof.
In particular, we have the following for the unweighted case.
where the minimum is over the same domain as in the previous corollary.
We now consider the mapping η given by η( Here I ∈ P f (IN ), the collection of finite sets of natural numbers, p = (p i ) i∈I with each p i > 0 and P I := i∈I p i , and x = (x i ) i∈I with each x i ∈ C.
Proof (i) Let p, q > 0 with P I , Q I > 0 (I ∈ P f (IN )). Then Since max{a, b} = (1/2)[a + b + |a − b|] for (a, b ∈ IR), we have from the definition of η that the last maximum in (1) can be written as For (ii), let I, J ∈ P f (IN ) with I ∩ J = ∅ and suppose p > 0 with P I , P J > 0. Then and we are done.
4 Two mappings associated with Jensen's inequality Suppose x i , y j ∈ C for i = 1, . . . , n and j = 1, . . . , m and θ i, In what follows, the mappings H, F : [0, 1] → IR are given by Proof Part (i) follows from Propositions 2.1 and 2.2 and part (ii) from the definition of F. The first inequality in (iii) derives from part (ii) and Lemma 2.3. The remainder of (iii) is a consequence of (ii) and the quasiconvexity of F.
In the special case m = 1 and y 1 = µ we write H = H 0 and F = F 0 . In the special case m = n and y i = x i (i = 1, . . . , n), we write H = H 1 and F = F 1 . These mappings were introduced by Dragomir in the case of f convex but have more general applicability. For notational convenience we rebadge the corresponding forms of θ i,j as Proof The outermost inequality of Theorem 3.1 may be written f (µ) ≤ max 1≤i≤n f (x i ), so by the definition of H 0 and the quasiconvexity of f whence we deduce the second inequality in (a).
The outermost inequality of Theorem 3.1 gives also that , whence the first inequality in (a).
From Theorem 3.1, we have successively 1], from which we have the rest of (b). Again by Theorem 3.1, , so (c) holds.

Further related maps
Some further maps on [0, 1] intimately related to H 0 , The mappings K and L were introduced (with different notation) in [1] and their properties studied in the case where f is convex. See also [2,3]. These mappings provided useful interpolations of Jensen's discrete inequality. Their behaviour in the convex context is similar to that of H 0 and F 1 respectively of the previous section. The present context is more subtle in that a sum of quasiconvex functions need not be quasiconvex.

Remark 5.1
We have from the definitions that for all t ∈ [0, 1] Proposition 5.2 For f quasiconvex
Also, by the definitions of quasiconvexity and K(t), which provides the first inequality in (3). For the remainder of (3), Theorem 3.1 provides f (x j ).

Proposition 5.3
For all t ∈ [0, 1], we have L(t) ≤ W (t) and Proof. The first inequality is provided by .
. . , n} and t ∈ [0, 1]. Multiplying by p i p j and summation over i, j yields which equals the right-hand side of the first inequality in (4).
which equals the right-hand side of the second inequality in (4). The proposition is proved. and Proof. From the definition of T , we have for t ∈ [0, 1] that whence the first inequality follows. Again by quasiconvexity Multiplying by p i and summation over i yields the second inequality. Similarly quasiconvexity supplies for all j ∈ {1, . . . , n} and t ∈ [0, 1] that Taking the maximum over j provides the third and final inequality.

Refinements of Jensen's inequality for quasiconvex functions
We begin by extending Theorem 3.1 to multisums. The following elementary lemma is useful.
Then the sequence (a k ) k≥1 is nonincreasing and bounded below by f (µ).
Proof. Take x i1,i2,...,i k = y 1,k in Lemma 6.2. The convexity of C ensures that x i1,i2,...,i k ∈ C. Then for each k ≥ 1, Lemma 6.2 gives Easy inductions on k provide so the left-hand side of (5) reduces to the required lower bound.
Put σ = x i (1 ≤ ≤ k +1) in Lemma 6.1 with K = k +1 and r i = 1/k for 1 ≤ i ≤ k and r k+1 = 0. We may extend the definition of y 1,k to y ,k for 1 ≤ ≤ k + 1 by setting y ,k = ρ . The condition By symmetry, each of the inner maxima takes the value max 1≤i1,...,i k ≤n {f (y 1,k )} = a k , so we have a k+1 ≤ a k , and we are done.
We may also derive a weighted refinement of Jensen's inequality for quasiconvex mappings.
Proof. We have just established the first inequality. For the second, take K = k in Lemma 6.1 with σ = x i and define r = q /Q k . We extend the definition of z 1,k to z ,k for 1 ≤ ≤ k by z ,k = ρ . Then K =1 r = 1 and so y 1,k = (1/k) Taking maxima provides Finally, by quasiconvexity f ( Proof. Quasiconvexity is immediate from Proposition 2.2. Now put by symmetry. This gives the first inequality in (6).
By Theorem 6.3, f (y 1,k ) ≤ max 1≤i≤n f (x i ), whence we derive the second inequality in (6). The remaining inequalities follow directly.
Define u k = u k (x i k+1 , . . . , x i 2k ) by u k = (x i k+1 + . . . + x i 2k )/k. A further sequence of mappings F [2k] n : [0, 1] → IR (k ≥ 1) associated with a quasiconvex f is given by where again each x i ∈ C (1 ≤ i ≤ n) and k ≥ 1. Proof. The proof follows familiar lines. We address only the pair of inequalities a k ≥ F which proves the first inequality. By the symmetry and quasiconvexity of F Proof. Quasiconvexity is immediate. By Lemma 6.2,