IJRRAS 16 (2) ● August 2013 www.arpapress.com/Volumes/Vol16Issue2/IJRRAS_16_2_04.pdf

GEOMETRY OF JENSEN’S INEQUALITY AND QUASI-ARITHMETIC MEANS

Zlatko Pavić Mechanical Engineering Faculty in Slavonski Brod, University of Osijek, Trg Ivane Brlić Mažuranić 2, 35000 Slavonski Brod, Croatia E-mail: [email protected]

ABSTRACT The paper observes the basic properties of convex function in the discrete and integral case. The connection of convexity with quantity centers is identified in the discrete case. The relation between barycenters and integral arithmetic means is studied in the integral case. The inequalities for quasi-arithmetic means are observed in both cases.

Key words and phrases: Convex function, chord, center of the convex combination, barycentre of the set, quasi-arithmetic mean. 2010 Mathematics Subject Classification: 26A51, 26E60, 28A10, 52A10.

1. INTRODUCTION Through this paper I   will be an interval with the non-empty interior I 0 . A characteristic function of a set X

will be denoted by  X , and convex hull of X will be denoted by coX . n n If x  I are points, and p [0,1] are coefficients such that p =1 , then the sum p x = c i i i=1 i i=1 i i belongs to I , and it is called the convex combination on I . The number c itself is called the center of the convex n combination. For a continuous function f : I   the associated convex combination p f (x ) belongs to i=1 i i f (I) .

cho Let f :[a,b]   be a convex function, and f[a,b] be the chord line joining the points A(a, f (a)) and B(b, f (b)) of the graph of f . Every x [a,b] can be presented as the convex combination b  x x  a x = a  b, b  a b  a and the convex function f satisfies the inequality b  x x  a f (x)  f (a)  f (b) = f cho (x). b  a b  a [a,b]

The polygonal chord line joining the points P1 (x1, f (x1 )),, Pn (xn , f (xn )) of the graph of f , so that

x1 << xn , will be used as the function n1 f plgcho (x) =  (x) f cho (x)   (x) f (x ). [x1,,xn ]  [xi ,xi1> [xi ,xi1] {xn} n i=1

If x[x1, xn ], then every convex function f :[x1, xn ]   verifies the double inequality f (x)  f plgcho (x)  f cho (x). (1) [x1,,xn ] [x1,xn ]

2. DISCRETE CASE The main result in this section is Theorem 2.5 which represents convexity by using the common center of different convex combinations. Let  be a plane. For a point P  the radius-vector OP considering some fixed point O will be   denoted by rP , and similarly, for a point Pi the radius-vector will be denoted by ri .

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Theorem A (Analytic Presentation of Convex Polygon) Let (P1,, Pn )   be a convex polygon with vertices P,, P . 1 n A radius-vector rP of a point P verifies the convex combination equality  n  (2) rP = pi ri i=1

if and only if P (P1,, Pn ) . Consequently,   n  n  (P1,, Pn ) = P  |rP = pi ri , pi [0,1],pi = 1. (3)  i=1 i=1 

Remark 2.1 Convex combinations in (2) are unique only for a line segment ( n = 2 ) and triangle ( n = 3 ). If we have a quadrangle (A, B,C, D) , then we can choose the point P that belongs to the triangles (A, B,C) and (A, B, D) , and does not belong to the edge AB . Using the radius-vectors of the vertices of these triangles, we have two different four-membered convex combinations so that          rP = pArA  pB rB  pC rC  0rD = qArA  qB rB  0rC  qDrD .

The above convex combinations are different because pC > 0 and qD > 0 .

Theorem B (Convex Polygon Presentation of Convex Function) A real valued function y = f (x) with real

variable x is (strictly) convex if and only if every n -tuple of the points Pi = Pi (xi , f (xi )) of the graph of f

delineates the convex polygon (with vertices Pi ).

Corollary 2.2 (Chord Presentation of Convex Function) A function f : I   is convex if and only if it satisfies the double inequality  n  n  n  f plgcho  p x   p f (x )  f cho  p x  (4) [x1,,xn ]  i i  i i [x1,xn ]  i i  i=1  i=1  i=1  n for every convex combination p x on I with x << x . i=1 i i 1 n n Proof. Suppose f is convex, and take any convex combination p x from I with x << x . Then the i=1 i i 1 n points P(x , f (x )) delineate the convex polygon  = coP ,, P  bounded with the chords f plgcho and i i i 1 n [x1,,xn ]   n  f cho by Theorem B. The end-point P of any radius-vector r of the form r = p r belongs to C by [x1,xn ] P P i=1 i i Theorem A. So we have the inclusion  n n  Ppi xi ,pi f (xi )   i=1 i=1  which represents the double inequality in (4). Suppose the inequality in (4) is valid, and take a binomial convex combination px  qy from I such that x < px  qy < y . Using the left hand side of the inequality in (4), we get plgcho f ( px  qy) = f[x, pxqy,y] ( px  qy)  pf (x)  qf (y) which proves the convexity of the function f .

The double inequality in (4) can be extended to the series of inequalities

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 n   n  n f  p x   f plgcho  p x   p f plgcho (x )  i i [x1,,xn ]  i i  i [x1,,xn ] i  i=1   i=1  i=1 n (5) = pi f (xi ) i=1 n  n   p f cho (x ) = f cho  p x .  i [x1,xn ] i [x1,xn ]  i i i=1  i=1  The second inequality in (5) is the well-known Jensen’s inequality which characterizes convex functions, that is, a function f is convex if and only if it satisfies Jensen’s inequality.

Figure 1: Graphical presentation of the inequality in (4)

We present briefly the physical meaning of convex combinations. Consider a set of n particles (points) in the plane. The value of a certain physical quantity q (mass, density, potential) is measured at each particle of the observed set.

We want to specify the center P of the quantity q . Let the particles be located at the points P1,, Pn with n non-negative quantity values q ,,q and positive quantity total value q = q . So, we can take the 1 n tot i=1 i  relative quantity values p = q /q for i =1,,n . It is reasonable to assume that the radius-vector r of the i i tot  P quantity center P is the convex combination of the given position vectors ri with the coefficients pi . Accordingly, we have  n  1 n  rP = pi ri = qi ri . (6) i=1 qtot i=1

Relying on Theorem A it can be concluded the center P is located in the convex hull of the points Pi , that is,

Pco{P1,, Pn}. Very practical descriptions of convexity was presented in [2].

Let x1,, xn I be points, and q1,,qn   be non-negative numbers such that k n 0 < q = q(k) < q(n) = q for some 1 k  n 1. If the convex combination equality i=1 i i=1 i 1 k 1 n qi xi = qi xi q(k) i=1 q(n) i=1 is valid, then it follows

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1 n 1  n k   qi xi = qi xi  qi xi  q(n)q(k) i=k1 q(n)q(k)  i=1 i=1  1  q(n) k k 

=  qi xi  qi xi  q(n)q(k)  q(k) i=1 i=1  1 k = qi xi . q(k) i=1 The next lemma gives the connection between quantity centers and Jensen’s type inequalities.

Lemma 2.3 Let x1,, xn I be points such that co{x1,, xk }{xk1,, xn} =  . Let k n q ,,q   be non-negative numbers such that 0 < q = q(k) < q(n) = q . 1 n i=1 i i=1 i If one of the convex combination equalities 1 k 1 n 1 n qi xi = qi xi = qi xi (7) q(k) i=1 q(n) i=1 q(n)q(k) i=k1 is valid, then the double convex combination inequality 1 k 1 n 1 n qi f (xi )  qi f (xi )  qi f (xi ) (8) q(k) i=1 q(n) i=1 q(n)q(k) i=k1 holds for every function f : I R which satisfies the double inequality in (4).

Proof. Let x stands for all convex combinations in (7). Without loss of generality suppose that all xi are pairwise

different, and x1 << xk . If we apply the right hand side of the inequality in (4) on the points x1,, xk , we get 1 k  1 k  q f (x )  f cho  q x  = f cho x .  i i [x1,xk ]  i i  [x1,xk ]   q(k) i=1  q(k) i=1 

Also suppose that {x1, xk , xk1,, xn} ={y1 << ynk2} . The point is in fact f cho x = f plgcho x . Applying the left hand side of the inequality in (4) on the points y ,, y , we [x1,xk ]   [ y1,,ynk2 ]   1 nk2 get  1 n  1 n f cho x = f plgcho  q x   q f (x ). [x1,xk ]   [ y1,,ynk2 ]   i i   i i  q(n)q(k) i=k1  q(n)q(k) i=k1 Connecting the above inequalities, we get 1 k 1 n qi f (xi )  qi f (xi ). q(k) i=1 q(n)q(k) i=k1 Then the binomial convex combination 1 n q(k)  1 k  q(n)q(k)  1 n      qi f (xi ) =  qi f (xi )   qi f (xi ) q(n) i=1 q(n)  q(k) i=1  q(n)  q(n)q(k) i=k1  represents the double inequality in (8).

Lemma 2.4 If a function f : I   satisfies the inequality in (8) under the condition in (7), then it satisfies the Jensen inequality. n Proof. Let x = p x . Without loss of generality suppose that all x are pairwise different, and all p > 0 . 0 i=1 i i i i

If x0  xi for all i , then we have co{x0}{x1,, xn} = . We can apply the outer side of the inequality in (8) n on the sets {x } and {x ,, x } to get the Jensen inequality (considering x = p x ) 0 1 n 0 i=1 i i

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n (9) f (x0 )  pi f (xi ). i=1 If x = x for some i = i , then the point x can be expressed again as the convex combination of the points 0 i0 0 0

different from x0 : 1 n

x0 = pi xi . 1 p i=1 i0 ii0

Since x0  xi for all i  i0 , we can apply the previous case and obtain the inequality 1 n

f (x0 )  pi f (xi ) 1 p i=1 i0 ii0 from which follows n (1 p ) f (x )  p f (x )  p f (x ). i0 0  i i i0 i0 i=1 Arranging the above inequality considering the assumption x = x , we get again the Jensen inequality in (9). i0 0

Theorem 2.5 (Quantity Center Presentation of Convex Function) A function f : I   is convex if and only if it satisfies the inequality in (8) under the condition in (7).

Example 2.6 Lemma 2.3 does not generally hold for convex functions of several variables as shown in the following

example in the coordinate plane. Take points P1(1,1) , P2 (1,1) , P3 (1,1) , P4 (1,1) , P5 (0,2) , P (0,2) , and function f (x, y) = x2 . Let C = coP, P , P , P . The square C does not contain the points 6  1 2 3 4 P and P . Let r be the radius-vectors of the points P . Then we have the double equality (all members equals 5 6 i i 0 ) 1 4  1 6  1 6  ri = ri = ri , 4 i=1 6 i=1 2 i=5 and strict double inequality 1 4 1 6 1 6

f (Pi ) > f (Pi ) > f (Pi ) 4 i=1 6 i=1 2 i=5 4 6 6 because f (P) = 4 , f (P) = 4 and f (P) = 0. i=1 i i=1 i i=5 i

In applications of convexity we often use strictly monotone continuous functions , : I   such that  is convex with respect to  ( is  -convex), that is, f =  1 is convex on (I) (this terminology is taken from [5, Definition 1.19]). A similar notation is used for concavity.

n Let p x be a convex combination on I . The discrete  -quasi-arithmetic mean of the points (particles) x i=1 i i i

with the coefficients (weights) pi is the point n 1    (xi , pi ) =  pi(xi ) (10)  i=1  n which belongs to I because the convex combination p (x ) belongs to (I) . i=1 i i

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Complete the section with the application of Lemma 2.3 on quasi-arithmetic means.

Theorem 2.7 Let x1,, xn I be points such that co{x1,, xk }{xk1,, xn} =  . Let k n q ,,q   be non-negative numbers such that 0 < q = q(k) < q(n) = q . Let , : I   1 n i=1 i i=1 i be strictly monotone continuous functions. If  is either  -convex and increasing or  -concave and decreasing, and if one of the equalities k n n  q   q   q   i   i   i  (11)   xi ,  =   xi ,  =   xi ,   q(k) i=1  q(n) i=1  q(n)q(k) i=k1 is valid, then the double inequality k n n  q   q   q   i   i   i  (12)   xi ,     xi ,     xi ,   q(k) i=1  q(n) i=1  q(n)q(k) i=k 1 holds. If  is either  -convex and decreasing or  -concave and increasing, then the reverse double inequality is valid in (12).

Proof. Briefly, if J = (I) we apply Lemma 2.3 with the points yi = (xi )  J and convex or concave function f =  1 : J   .

More generally about the different forms of quasi-arithmetic means can be found in the article [1].

3. INTEGRAL CASE The main result in this section is Theorem 3.5 which represents convexity by using the common barycenter of different intervals. The integral variants of Jensen’s type inequalities are also obtained by using the barycenters.

We will use a finite measure  on I assuming that all subintervals of I are  -measurable.

Integral analogy of the concept of convex combination is the concept of barycenter. Let A  I be a  -measurable n with (A) > 0. Given a positive integer n , let A = i=1 Ani be a partition of pairwise disjoint  -measurable

sets Ani , and xni  Ani be points. Then we have the convex combination n (Ani )  xni = cn i=1 (A)

whose center cn belongs to coA . If the sequence (cn )n converges, then the  -barycenter of A can be defined by n  (Ani )  1 (A, ) = lim  xni  = xd(x). (13)   A n  i=1 (A)  (A) Integral arithmetic mean of a function is similarly defined. If a function f : I   is  -integrable on A , then the  -arithmetic mean of f on A is defined by n  (Ani )  1 ( f , A, ) = lim  f (xni ) = f (x)d(x). (14)   A n  i=1 (A)  (A)

Note that (1A , A, ) = (A, ) where 1A denotes the identity function on A . If A is the interval, then its  -barycenter (A, ) belongs to A , and if f is continuous on A , then its  -arithmetic mean on A belongs to f (A) .

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The basic rule for summation of barycenters and integral arithmetic means says:

n Proposition 1 Let  be a finite measure on I , and A = i=1 Ai be a union of pairwise disjoint  -measurable

sets Ai  I with (Ai ) > 0 . If the  -barycenter (A, ) exists, then 1 n (A, ) = (Ai )(Ai , ). (15) (A) i=1

If a function f : I   is  -integrable on A , then 1 n ( f , A, ) = (Ai )( f , Ai , ). (16) (A) i=1 Integral analogy of Corollary 2.2 is one variant of the Hermite-Hadamard inequality which shows the next corollary.

Corollary 3.1 Let  be a finite measure on I , and f : I   be a function. If f is convex and continuous, then it satisfies the double inequality 1 f ( pa  qb)  f (x)d(x)  pf (a)  qf (b) (17) ([a,b]) [a,b] for every bounded closed interval [a,b]  I with ([a,b]) > 0 , and its  -barycenter (A,) = (1/([a,b])) xd(x) = pa  qb . [a,b] n Proof. Given a positive integer n , let [a,b] = i=1 Ani be a partition of pairwise disjoint intervals Ani where

every Ani contracts to the point as n goes to infinity. For every i =1,,n we take one point xni  Ani , and

also take pni = (Ani)/([a,b]) . Suppose that xn1 << xnn . Applying the inequality in (4) for this case, we have  n  n  n  f plgcho  p x   p f (x )  f cho  p x . (18) [xn1,,xnn]  ni ni  ni ni [xn1,xnn]  ni ni  i=1  i=1  i=1  The following limits  n  1 lim  pnixni  = xd(x) = pa  qb  [a,b] n  i=1  ([a,b])   n  lim  f plgcho  p x  = f ( pa  qb)  [xn1,,xnn]  ni ni  n  i=1     n  1 lim  pni f (xni ) = f (x)d(x)  [a,b] n  i=1  ([a,b])   n  lim  f cho  p x  = f cho ( pa  qb) = pf (a)  qf (b)  [xn1,xnn]  ni ni  [a,b] n   i=1  hold, so the inequality in (18) "converges" to the inequality in (17).

Basic connections between the convexity and the Hermite-Hadamard inequality can be found in [3, pages 50-53].

Now wee will use the finite measure  on I which is positive on the intervals, that is, which satisfies (A) > 0 for every non-degenerate interval A  I .

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Using the integral method with convex combinations as in the previous corollary, we get the integral variant of Lemma 2.3:

Lemma 3.2 Let  be a finite measure on I that is positive on intervals. Let A, B  I be intervals so that A  B and (A) < (B) . If one of the barycenter equalities (A,) = (B,) = (B\A,) (19) is valid, then the double integral arithmetic mean inequality ( f , A, )  ( f , B, )  ( f , B\A, ) (20) holds for every  -integrable convex function f : I   . n k Proof. Given a positive integer n , let B = i=1 Ani be a partition, with the subpartition A = i=1 Ani , of pairwise

disjoint intervals Ani where every Ani contracts to the point or vanishes in infinity as n goes to infinity. Put

qni = (Ani) . Since we suppose the barycenter equality in (19) is valid, we can choose the points xni  Ani that satisfy the center equality 1 k 1 n 1 n

qnixni = qnixni = qnixni. (A) i=1 (B) i=1 (B \ A) i=k1

For example, we can put xni = (Ani,) = (1/(Ani )) xd(x) . So we can apply the inequality in (8), and Ani get 1 k 1 n 1 n

qni f (xni )  qni f (xni )  qni f (xni ). (A) i=1 (B) i=1 (B \ A) i=k1 Letting n to infinity, we obtain the inequality in (20).

The variant of Lemma 3.2 for the bounded closed intervals A and B was proved in [4, Proposition 1] by using the cho chord line y = f[a,b] (x) in the case A = [a,b] . Lemma 3.2 is not generally true for convex functions of several variables as can be seen in [4, Examples 1 and 2].

Remark 3.3 Lemma 3.2 is also valid in the case when the set B is a union of intervals.

A measure  on I is said to be continuous if ({t}) = 0 for every point t  I (according to the definition in the book [6, page 149]).

Lemma 3.4 Let  be a continuous finite measure on I that is positive on the intervals. 0 If a I is a point, then the decreasing series (An )n of bounded intervals An  I exists so that  (An , ) = a and An = {a}. n=1 Proof. Take a point a I 0 , and show the first two steps.

In the first step, we choose the points x1, y1 I such that x1 < a < y1 , and determine the  -barycenter of the

interval [x1, y1] : 1 a1 = ([x1, y1 ], ) = td(t). [x ,y ] [x1, y1 ] 1 1

If a1 = a , then we take A1 = [x1, y1]. If a1 > a , then we observe the function g :[x1, y1 ]   defined by 1 g(x) = ([x1, x], )  a = td(t)  a. [x ,x] ([x1, x]) 1

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The function g is strictly increasing continuous with g(x1) = x1  a < 0 and g(y1) = a1  a > 0 . Therefore,

the function g has the unique zero-point y1  (in fact y1  ). In this case, we take

A1 = [x1, y1] . If a1 < a , we use the function g(x) = ([x, y1 ], )  a .

In the next step, if A1 = [x1, y1], we take the points x  a a  y x = 1 and y = 1 , 2 2 2 2

and repeat the previous procedure to determine A2 .

Theorem 3.5 (Barycenter Presentation of Convex Function) Let  be a continuous finite measure on I that is positive on the intervals. A  -integrable continuous function f : I   is convex if and only if it satisfies the inequality in (20) under the condition in (19).

Proof. Necessity is proved in Lemma 3.2. Now let us prove the sufficiency. Take any convex combination px  qy = a from I with x < y and p,q<0,1> . Then a I 0 , so we can

apply Lemma 3.4 to determine the decreasing series (An )n of bounded intervals An which contracts to {a} , and

every barycenter (An , ) = a . In a similar way, for every n it is possible to determine the union

[x, xn ][yn , y] so that ([x, xn ][yn , y], ) = a , and that xn  x and yn  y as n   . Let

Bn = [x, xn ] An [yn , y] , and

([x, xn ]) ([yn , y]) pn = ,qn = . ([x, xn ])  ([yn , y]) ([x, xn ])  ([yn , y]) Then the equality 1 p q td(t) = n td(t)  n td(t), B \ A [x,x ] [ y ,y] (Bn \ An ) n n ([x, xn ]) n ([yn , y]) n respectively

a = pn ([x, xn ], )  qn ([yn , y], )

holds for every n . Since lim n ([x, xn ], ) = x and lim n ([yn , y], ) = y , then lim n pn = p

and lim n qn = q .

Applying the outer side of the inequality in (20) on the pair of the sets An and Bn , we get 1 1 f (t)d(t)  f (t)d(t) (A ) An (B \ A ) Bn \ An n n n p q = n f (t)d(t)  n f (t)d(t). [x,x ] [ y , y] ([x, xn ]) n ([yn , y]) n Letting n  , and using the continuity of f , we achieve f (a)  pf (x)  qf (y) which proves the convexity of f because a = px  qy .

We need the following generalization of Lemma 3.2 for applications on quasi-arithmetic means.

Corollary 3.6 Let  be a finite measure on I that is positive on the intervals. Let A, B  I be intervals so that A  B and (A) < (B) . Let g : I   be a  -integrable continuous function, and J = g(I) . If one of the integral arithmetic mean equalities

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IJRRAS 16 (2) ● August 2013 Pavić ● Geometry of Jensen’s Inequality

(g, A, ) = (g, B, ) = (g, B\A, ) (21) is valid, then the double integral arithmetic mean inequality ( f  g, A, )  ( f  g, B, )  ( f  g, B\A, ) (22) holds for every convex function f : J   provided that f  g is  -integrable.

Let  be a finite measure on I , A  I be  -measurable set with (A) > 0, and  : I   be a strictly monotone continuous function that is  -integrable on A . The integral  -quasi-arithmetic mean on the set A with respect to the measure  is the number

1  1   A,  =   (x)d(x). (23)  A   (A) 

If A is the interval, then  A,  belongs to A because (1/(A)) (x)d(x) belongs to (A) . If A

A is not connected, then  A,  may be outside of A . If 1I denotes the identity function on I , then

1I -quasi-arithmetic mean on A is just the  -barycenter of A , that is,

1 (A, ) = (1/(A)) xd(x) = (A, ) . I A

Theorem 3.7 Let  be a finite measure on I that is positive on the intervals. Let A, B  I be intervals so that A  B and (A) < (B) . Let , : I   be  -integrable strictly monotone continuous functions. If  is either  -convex and increasing or  -concave and decreasing, and if one of the equalities

 (A, ) =  (B, ) =  (B\A, ) (24) is valid, then the double inequality

 (A, )   (B, )   (B\A, ) (25) holds. If  is either  -convex and decreasing or  -concave and increasing, then the reverse double inequality is valid in (25). Proof. Let us prove the case when  is  -convex and increasing. First we apply the function  on the equalities in (24), and get 1 1 1 (x)d(x) = (x)d(x) = (x)d(x). (A) A (B) B (B\A) B\ A Put J = (I) . Now, we can apply Corollary 3.6 with convex function f =  1 : J R , and since f ((x)) = (x) , we have 1 1 1  (x)d(x)   (x)d(x)   (x)d(x). (A) A (B) B (B\A) B\ A Finally, we apply the increasing function  1 on the above double inequality, and get the double inequality in (25).

References  [1] J. Mi c i c , Z. Pavi c , and J. Pe c ari c , "The inequalities for quasiarithmetic means", Abstract and Applied Analysis, vol. 2012, Article ID 203145, 25 pages, 2012. [2] T. Needham, "A visual explanation of Jensen’s inequality", American Mathematical Monthly, vol. 100, pp. 768-771, 1993. [3] C. P. Niculescu, and L. E. Persson, Convex Functions and Their Applications, Canadian Mathematical Society, Springer, New York, USA, 2006.  [4] Z. Pavi c , J. Pe c ari c , and I. Peri c , "Integral, discrete and functional variants of Jensen’s inequality", Journal of Mathematical Inequalities, vol. 5, no. 2, pp. 253-264, 2011. [5] J. E. Pe c ari c , F. Proschan, and Y. L. Tong, Convex Functions, Partial Orderings, and Statistical Applications, Academic Press, New York, USA, 1992. [6] W. Rudin, Real and Complex Analysis, McGraw-Hill, New York, USA, 1987.

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