Some inequalities of the Edmundson-Lah-Ribaric type for 3-convex functions with applications
aa r X i v : . [ m a t h . G M ] J a n SOME INEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČTYPE FOR 3-CONVEX FUNCTIONS WITH APPLICATIONS
ROZARIJA MIKIĆ, ÐILDA PEČARIĆ, AND JOSIP PEČARIĆ
Abstract.
In this paper we derive some Edmundson-Lah-Ribarič type in-equalities for positive linear functionals and 3-convex functions. Main re-sults are applied to the generalized f -divergence functional. Examples withZipf–Mandelbrot law are used to illustrate the results. In addition, obtainedresults are utilized in constructing some families of exponentially convex func-tions and Stolarsky-type means. Introduction
The importance of Jensen’s inequality for convex functions is its applicability invarious branches of mathematics, especially in mathematical analysis and statis-tics. In this paper we refer to a general form of the Jensen inequality for positivelinear functionals. In order to present our result, we first need to introduce theappropriate setting.Let E be a non-empty set and let L be a vector space of real-valued functions f : E → R having the properties:(L1) f, g ∈ L ⇒ ( af + bg ) ∈ L for all a, b ∈ R ;(L2) ∈ L , i.e., if f ( t ) = 1 for every t ∈ E , then f ∈ L .We also consider positive linear functionals A : L → R . That is, we assumethat:(A1) A ( af + bg ) = aA ( f ) + bA ( g ) for f, g ∈ L and a, b ∈ R ;(A2) f ∈ L , f ( t ) ≥ for every t ∈ E ⇒ A ( f ) ≥ ( A is positive).Since it was proved, the famous Jensen inequality and its converses have beenextensively studied by many authors and have been generalized in numerousdirections. Jessen [19] gave the following generalization of Jensen’s inequality forconvex functions (see also [29, p.47]): Theorem 1.1. ( [19] ) Let L satisfy properties (L1) and (L2) on a nonempty set E , and assume that φ is a continous convex function on an interval I ⊂ R . If Mathematics Subject Classification.
Key words and phrases.
Jensen inequality, Edmundson-Lah-Ribarič inequality, 3-convexfunctions, f -divergence, Zipf-Mandelbrot law, exponential convexity, Stolarsky-type means. A is a positive linear functional with A (1) = 1 , then for all f ∈ L such that φ ( f ) ∈ L we have A ( f ) ∈ I and φ ( A ( f )) ≤ A ( φ ( f )) . (1.1)The following result is one of the most famous converses of the Jensen inequalityknown as the Edmundson-Lah-Ribarič inequality, and it was proved in [2] byBeesack and Pečarić (see also [29, p.98]): Theorem 1.2. ( [2] ) Let φ be convex on the interval I = [ m, M ] such that −∞ Let a function f be defined on an interval I ⊆ R . The followingequivalences hold. (i) If f ∈ C ( I ) , then f is 3-convex if and only if [ t, t, t , t ] f ≥ for allmutually different points t, t , t ∈ I . (ii) If f ∈ C ( I ) , then f is 3-convex if and only if [ t, t, t , t ] f ≥ for allmutually different points t, t ∈ I . (iii) If f ∈ C ( I ) , then f is 3-convex if and only if [ t, t, t, t ] f ≥ for allmutually different points t, t ∈ I . (iv) If f ∈ C ( I ) , then f is 3-convex if and only if [ t, t, t, t ] f ≥ for every t ∈ I . Results Throughout this paper, whenever mentioning the interval [ m, M ] , we assumethat −∞ < m < M < ∞ holds. Theorem 2.1. Let L satisfy conditions (L1) and (L2) on a non-empty set E and let A be any positive linear functional on L with A ( ) = 1 . Let φ be a 3-convex function defined on an interval of real numbers I whose interior contains R. MIKIĆ, Ð. PEČARIĆ, AND J. PEČARIĆ the interval [ m, M ] .Then A [( M − f )( f − m )] M − m (cid:18) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) (cid:19) ≤ M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) (2.1) ≤ A [( M − f )( f − m )] M − m (cid:18) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m (cid:19) holds for any f ∈ L such that φ ◦ f ∈ L and m ≤ f ( t ) ≤ M for t ∈ E . If thefunction − φ is 3-convex, then the inequalities are reversed.Proof: The function φ is 3-convex, so from Theorem 1.3 (i) we have that [ t, t, t , t ] φ ≥ for all mutually different points t, t , t ∈ I . When we take t = m , t = x and t = M in (1.3), we obtain that ≤ φ ′ + ( m )( m − x )( m − M ) + φ ( m )( x + M − m )( m − x ) ( m − M ) + φ ( x )( x − m ) ( x − M ) + φ ( M )( M − m ) ( M − x ) holds for every x ∈ h m, M i . After multiplying by ( x − m ) ( x − M ) and rearranging,the upper relation becomes ( M − x )( x − m ) M − m (cid:18) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) (cid:19) ≤ M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) . (2.2)In a similar manner, when we put t = M , t = x and t = m in (1.3), afterarranging the relation thus obtained, we get that M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) ≤ ( M − x )( x − m ) M − m (cid:18) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m (cid:19) (2.3)holds for every x ∈ h m, M i . Now, we see that (2.2) and (2.3) give the followingsequence of inequalities: ( M − x )( x − m ) M − m (cid:18) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) (cid:19) ≤ M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) ≤ ( M − x )( x − m ) M − m (cid:18) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m (cid:19) . (2.4) NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 5 Since the function f ∈ L satisfies the bounds m ≤ f ( t ) ≤ M , we can replace x with f ( t ) in (2.4), and get ( M − f ( t ))( f ( t ) − m ) M − m (cid:18) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) (cid:19) ≤ M − f ( t ) M − m φ ( m ) + f ( t ) − mM − m φ ( M ) − φ ( f ( t )) ≤ ( M − f ( t ))( f ( t ) − m ) M − m (cid:18) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m (cid:19) . The inequalities (2.1) follow after applying linear functional A to the previousrelation taking into account linearity of the functional A and condition A ( ) = 1 . ✷ Remark 2.1. The result from Theorem 2.1 is already proven in the paper [26] ,but in this paper we have provided a shorter and more elegant proof. Theorem 2.2. Let L satisfy conditions (L1) and (L2) on a non-empty set E and let A be any positive linear functional on L with A ( ) = 1 . Let φ be a 3-convex function defined on an interval of real numbers I whose interior containsthe interval [ m, M ] and differentiable on h m, M i . Then ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m )2 (cid:21) − A [( f − m ) φ ′ ( f )] ≤ M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) (2.5) ≤ A [( M − f ) φ ′ ( f )] − ( M − A ( f )) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′− ( M )2 (cid:21) holds for any f ∈ L such that φ ◦ f ∈ L and m ≤ f ( t ) ≤ M for t ∈ E . If thefunction − φ is 3-convex, then the inequalities are reversed.Proof: Let φ be a 3-convex function. From Theorem 1.3 (ii) we have that [ t, t, t , t ] φ ≥ for all mutually different points t, t ∈ I . When we take t = m and t = x in (1.4), we obtain that ≤ x − m ) (cid:2) ( x − m )( φ ′ ( x ) + φ ′ + ( m )) + 2( φ ( m ) − φ ( x )) (cid:3) holds for every x ∈ h m, M i . After multiplying by ( x − m ) and rearranging, therelation from above becomes ( x − m ) (cid:20) φ ( M ) − φ ( m ) M − m − (cid:0) φ ′ ( x ) + φ ′ + ( m ) (cid:1)(cid:21) ≤ M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) . (2.6) R. MIKIĆ, Ð. PEČARIĆ, AND J. PEČARIĆ Similarly, when we put t = M and t = x in (1.4) and rearrange the obtainedrelation, we get that M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) ≤ ( M − x ) (cid:20) (cid:0) φ ′ ( x ) + φ ′− ( M ) (cid:1) − φ ( M ) − φ ( m ) M − m (cid:21) (2.7)holds for every x ∈ h m, M i . Now, we see that (2.6) and (2.7) together give thefollowing sequence of inequalities: ( x − m ) (cid:20) φ ( M ) − φ ( m ) M − m − (cid:0) φ ′ ( x ) + φ ′ + ( m ) (cid:1)(cid:21) ≤ M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) ≤ ( M − x ) (cid:20) (cid:0) φ ′ ( x ) + φ ′− ( M ) (cid:1) − φ ( M ) − φ ( m ) M − m (cid:21) . (2.8)Since the function f ∈ L satisfies the bounds m ≤ f ( t ) ≤ M , we can replace x with f ( t ) in (2.8), and get ( f ( t ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − (cid:0) φ ′ ( f ( t )) + φ ′ + ( m ) (cid:1)(cid:21) ≤ M − f ( t ) M − m φ ( m ) + f ( t ) − mM − m φ ( M ) − φ ( f ( t )) ≤ ( M − f ( t )) (cid:20) (cid:0) φ ′ ( f ( t )) + φ ′− ( M ) (cid:1) − φ ( M ) − φ ( m ) M − m (cid:21) . The inequalities (2.5) follow after applying linear functional A to the previousrelation taking into account linearity of the functional A and condition A ( ) = 1 . ✷ Remark 2.2. If it exists, the first derivative φ ′ of a 3-convex function φ is aconvex function. It is known that convex functions are continuous on every openinterval, and their one-sided derivatives exist and are finite. Theorem 2.3. Let L satisfy conditions (L1) and (L2) on a non-empty set E and let A be any positive linear functional on L with A ( ) = 1 . Let φ be a 3-convex function defined on an interval of real numbers I whose interior contains NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 7 the interval [ m, M ] and differentiable on h m, M i . Then ( M − A ( f )) (cid:20) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m (cid:21) − φ ′′− ( M )2 A [( M − f ) ] ≤ M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) (2.9) ≤ ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) (cid:21) − φ ′′ + ( m )2 A [( f − m ) ] holds for any f ∈ L such that φ ◦ f ∈ L and m ≤ f ( t ) ≤ M for t ∈ E . If thefunction − φ is 3-convex, then the inequalities are reversed.Proof: The function φ is 3-convex on [ m, M ] and twice differentiable, so fromTheorem 1.3 (iii) we have that [ t, t, t , t ] φ ≥ for all mutually different points t, t, t ∈ [ m, M ] . When we take t = m and t = x in (1.5), we obtain that ≤ x − m ) " φ ( x ) − X k =0 φ ( k )+ ( m ) k ! ( x − m ) k holds for every x ∈ h m, M i . After multiplying by ( x − m ) and rearranging, theupper relation becomes M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) ≤ ( x − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) − φ ′′ + ( m )2 ( x − m ) (cid:21) . (2.10)In a similar manner, when we put t = M and t = x in (1.5), after rearrangingthe relation thus obtained, we get that ( M − x ) (cid:20) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m − φ ′′− ( M )2 ( M − x ) (cid:21) ≤ M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) (2.11)holds for every x ∈ h m, M i . Now, we see that (2.10) and (2.11) give the followingsequence of inequalities: ( M − x ) (cid:20) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m − φ ′′− ( M )2 ( M − x ) (cid:21) ≤ M − xM − m φ ( m ) + x − mM − m φ ( M ) − φ ( x ) ≤ ( x − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) − φ ′′ + ( m )2 ( x − m ) (cid:21) . (2.12) R. MIKIĆ, Ð. PEČARIĆ, AND J. PEČARIĆ Since the function f ∈ L satisfies the bounds m ≤ f ( t ) ≤ M , we can replace x with f ( t ) in (2.12), and get ( M − f ( t )) (cid:20) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m − φ ′′− ( M )2 ( M − f ( t )) (cid:21) ≤ M − f ( t ) M − m φ ( m ) + f ( t ) − mM − m φ ( M ) − φ ( f ( t )) ≤ ( f ( t ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) − φ ′′ + ( m )2 ( f ( t ) − m ) (cid:21) . The inequalities (2.9) follow after applying linear functional A to the previousrelation taking into account linearity of the functional A and condition A ( ) = 1 . ✷ Remark 2.3. Theorems 2.2 and 2.3 can be utilized for obtaining following Jensen-type inequalities for 3-convex functions. (i) When we put x = A ( f ) in scalar inequalities (2.8) and then subtract theinequalities from Theorem 2.2, we get ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − (cid:0) φ ′ ( A ( f )) + φ ′ + ( m ) (cid:1)(cid:21) − A [( M − f ) φ ′ ( f )] − ( M − A ( f )) (cid:20) φ ( M ) − φ ( m ) M − m + φ ′− ( M )2 (cid:21) ≤ A ( φ ( f )) − φ ( A ( f )) ≤ ( M − A ( f )) (cid:20) (cid:0) φ ′ ( A ( f )) + φ ′− ( M ) (cid:1) − φ ( M ) − φ ( m ) M − m (cid:21) − ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m )2 (cid:21) + 12 A [( f − m ) φ ′ ( f )] . (ii) When we put x = A ( f ) in scalar inequalities (2.12) and then subtract theinequalities from Theorem 2.3, we get ( M − A ( f )) (cid:20) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m − φ ′′− ( M )2 ( M − A ( f )) (cid:21) − ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) (cid:21) + φ ′′ + ( m )2 A [( f − m ) ] ≤ A ( φ ( f )) − φ ( A ( f )) ≤ ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m ) − φ ′′ + ( m )2 ( A ( f ) − m ) (cid:21) − ( M − A ( f )) (cid:20) φ ′− ( M ) − φ ( M ) − φ ( m ) M − m (cid:21) + φ ′′− ( M )2 A [( M − f ) ] . Remark 2.4. Theorem 2.1 can likewise be used to obtain Jensen-type inequalitiesfor 3-convex functions, and that result is already given in [26] . NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 9 Applications to Csiszár divergence and Zipf-Mandelbrot law Let us denote the set of all probability distributions by P , that is we say p = ( p , ..., p n ) ∈ P if p i ∈ [0 , for i = 1 , ..., n and P ni =1 p i = 1 .Numerous theoretic divergence measures between two probability distributionshave been introduced and comprehensively studied. Their applications can befound in the analysis of contingency tables [11], in approximation of probabilitydistributions [6], [24], in signal processing [20], and in pattern recognition [3], [5].Csiszár [7]-[8] introduced the f − divergence functional as D f ( p , q ) = n X i =1 q i f (cid:18) p i q i (cid:19) , (3.1)where f : [0 , + ∞i is a convex function, and it represent a "distance function" onthe set of probability distributions P .A great number of theoretic divergences are special cases of Csiszár f -divergencefor different choices of the function f .As in Csiszár [8], we interpret undefined expressions by f (0) = lim t → + f ( t ) , · f (cid:18) (cid:19) = 0 , · f (cid:16) a (cid:17) = lim ǫ → + f (cid:16) aǫ (cid:17) = a · lim t →∞ f ( t ) t . In this section we will study a generalization of the f -divergence functionalfor the class of 3-convex functions. It is an extension of the results obtained in[26]. Throughout this section, when mentioning the interval [ m, M ] , we assumethat [ m, M ] ⊆ R + . For a 3-convex function f : [ m, M ] → R we give the followingdefinition of generalized f -divergence functional found in [26]: ˜ D f ( p , q ) = n X i =1 q i f (cid:18) p i q i (cid:19) . (3.2)We can utilize Theorem 2.2 to get an Edmundson-Lah-Ribarič type inequalityfor the above defined generalized f -divergence functional. Theorem 3.1. Let [ m, M ] ⊂ R be an interval such that m ≤ ≤ M . Let f be a 3-convex function on the interval I whose interior contains [ m, M ] anddifferentiable on h m, M i . Let p = ( p , ..., p n ) and p = ( q , ..., q n ) be probability distributions such that p i /q i ∈ [ m, M ] for every i = 1 , ..., n . Then we have (1 − m ) (cid:20) f ( M ) − f ( m ) M − m − f ′ + ( m )2 (cid:21) − n X i =1 ( p i − mq i ) f ′ (cid:18) p i q i (cid:19) ≤ M − M − m f ( m ) + 1 − mM − m f ( M ) − ˜ D f ( p , q ) (3.3) ≤ n X i =1 ( M q i − p i ) f ′ (cid:18) p i q i (cid:19) − ( M − (cid:20) f ( M ) − f ( m ) M − m − f ′− ( M )2 (cid:21) . Proof: Let x = ( x , ..., x n ) such that x i ∈ [ m, M ] for i = 1 , ..., n . Let φ be a 3-convex function on the interval I whose interior contains [ m, M ] and differentiableon h m, M i . In the relation (2.5) we can replace f ←→ x , and A ( x ) = n X i =1 p i x i . In that way we get (¯ x − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ + ( m )2 (cid:21) − n X i =1 p i ( x i − m ) φ ′ ( x i ) ≤ M − ¯ xM − m φ ( m ) + ¯ x − mM − m φ ( M ) − n X i =1 p i φ ( x i ) ≤ n X i =1 p i ( M − x i ) φ ′ ( x i ) − ( M − ¯ x ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′− ( M )2 (cid:21) where ¯ x = P ni =1 p i x i . Since the function f satisfies the same assumtions as φ , inthe previous relation we can set φ = f, p i = q i and x i = p i q i , and after calculating ¯ x = n X i =1 q i p i q i = n X i =1 p i = 1 we get (3.3). ✷ By utilizing Theorem 2.3 in the analogous way as above, we get a differentEdmundson-Lah-Ribarič type inequality for the generalized f -divergence func-tional (3.2), and it is given in the following theorem. Theorem 3.2. Let [ m, M ] ⊂ R be an interval such that m ≤ ≤ M . Let f be a 3-convex function on the interval I whose interior contains [ m, M ] and NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 11 differentiable on h m, M i . Let p = ( p , ..., p n ) and p = ( q , ..., q n ) be probabilitydistributions such that p i /q i ∈ [ m, M ] for every i = 1 , ..., n . Then we have ( M − (cid:20) f ′− ( M ) − f ( M ) − f ( m ) M − m (cid:21) − f ′′− ( M )2 n X i =1 ( M q i − p i ) q i ≤ M − M − m f ( m ) + 1 − mM − m f ( M ) − ˜ D f ( p , q ) (3.4) ≤ (1 − m ) (cid:20) f ( M ) − f ( m ) M − m − f ′ + ( m ) (cid:21) − f ′′ + ( m )2 n X i =1 ( p i − mq i ) q i . Remark 3.1. Theorem 2.1 can be in analogue way applied to generalized Csiszárdivergence functional, but since this application is already shown in [26] , we omitit. Example 3.1. Let p = ( p , ..., p n ) and p = ( q , ..., q n ) be probability distributionsand let [ m, M ] ⊂ R be an interval such that m ≤ ≤ M and p i /q i ∈ [ m, M ] forevery i = 1 , ..., n . ⊲ Kullback-Leibler divergence of the probability distributions p and q isdefined as D KL ( p , q ) = n X i =1 q i log q i p i , and the corresponding generating function is f ( t ) = t log t, t > . We cancalculate f ′′′ ( t ) = − t < , so the function − f ( t ) = − t log t is 3-convex.Now it is obvious that for the Kullback-Leibler divergence the inequalities(3.3) and (3.4) hold with reversed signs of inequality, with f ′ + ( m ) = log m + 1 , f ′− ( M ) = log M + 1 and f ′′ + ( m ) = 1 m , f ′′− ( M ) = 1 M .⊲ Hellinger divergence of the probability distributions p and q is definedas D H ( p , q ) = 12 n X i =1 ( √ q i − √ p i ) , and the corresponding generating function is f ( t ) = (1 − √ t ) , t > .We see that f ′′′ ( t ) = − t − < , so the function − f ( t ) = − (1 − √ t ) is 3-convex. It is clear that for the Hellinger divergence the inequalities(3.3) and (3.4) hold with reversed signs of inequality, with f ′ + ( m ) = − √ m + 12 , f ′− ( M ) = − √ M + 12 and f ′′ + ( m ) = 14 √ m , f ′′− ( M ) = 14 √ M .⊲ Renyi divergence of the probability distributions p and q is defined as D α ( p , q ) = n X i =1 q α − i p αi , α ∈ R , and the corresponding generating function is f ( t ) = t α , t > . We calculatethat f ′′′ ( t ) = α ( α − α − t α − and see that the function f ( t ) = t α is3-convex for ≤ α ≤ and α ≥ , and − f ( t ) = − t α is 3-convex for α ≤ and < α < , and we have f ′ + ( m ) = αm α − , f ′− ( M ) = αM α − ,f ′′ + ( m ) = α ( α − m α − and f ′′− ( M ) = α ( α − M α − . As regards, the Renyi divergence, the inequalities (3.3) and (3.4) hold for ≤ α ≤ and α ≥ , and if α ≤ or < α < the signs of inequalityare reversed. ⊲ Harmonic divergence of the probability distributions p and q is definedas D Ha ( p , q ) = n X i =1 p i q i p i + q i , and the corresponding generating function is f ( t ) = t t . We can calculate f ′′′ ( t ) = t ) > , so the function f is 3-convex. Now it is obvious thatfor the harmonic divergence the inequalities (3.3) and (3.4) hold with f ′ + ( m ) = 2(1 + m ) , f ′− ( M ) = 2(1 + M ) and f ′′ + ( m ) = − m ) , f ′′− ( M ) = − M ) .⊲ Jeffreys divergence of the probability distributions p and q is defined as D J ( p , q ) = 12 n X i =1 ( q i − p i ) log q i p i , and the corresponding generating function is f ( t ) = (1 − t ) log t , t > .We see that f ′′′ ( t ) = − t − t < , so the function − f ( t ) = (1 − t ) log t is3-convex. Instantly we get that for the Jeffreys divergence the inequalities(3.3) and (3.4) hold with reversed signs of inequality, with f ′ + ( m ) = log m − m + 1 , f ′− ( M ) = log M − M + 1 NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 13 and f ′′ + ( m ) = 1 m + 1 m , f ′′− ( M ) = 1 M + 1 M . Examples with Zipf and Zipf-Mandelbrot law Zipf’s law [32], [33] has and continues to attract considerable attention in awide variety of scientific disciplines - from astronomy to demographics to softwarestructure to economics to zoology, and even to warfare [10]. It is one of thebasic laws in information science and bibliometrics, but it is also often usedin linguistics. Same law in mathematical sense is also used in other scientificdisciplines, but name of the law can be different, since regularities in differentscientific fields are discovered independently from each other. Typically one isdealing with integer-valued observables (numbers of objects, people, cities, words,animals, corpses) and the frequency of their occurrence.Probability mass function of Zipf’s law with parameters N ∈ N and s > is: f ( k ; N, s ) = 1 /k s H N,s , where H N,s = N X i =1 i s . Benoit Mandelbrot in 1966 gave an improvement of Zipf law for the count ofthe low-rank words. Various scientific fields use this law for different purposes,for example information sciences use it for indexing [9, 31], ecological field studiesin predictability of ecosystem [27], in music it is used to determine aestheticallypleasing music [25].Zipf–Mandelbrot law is a discrete probability distribution with parameters N ∈ N , q, s ∈ R such that q ≥ and s > , possible values { , , ..., N } and probabilitymass function f ( i ; N, q, s ) = 1 / ( i + q ) s H N,q,s , where H N,q,s = N X i =1 i + q ) s . (4.1)Let p and q be Zipf-Mandelbrot laws with parameters N ∈ N , q , q ≥ and s , s > respectively and let us denote m p , q := min (cid:26) p i q i (cid:27) = H N,q ,s H N,q ,s min (cid:26) ( i + q ) s ( i + q ) s (cid:27) M p , q := max (cid:26) p i q i (cid:27) = H N,q ,s H N,q ,s max (cid:26) ( i + q ) s ( i + q ) s (cid:27) (4.2)In this section we utilize the results regarding Csiszár divergence from theprevious section in order to obtain different inequalities for the Zipf-Mandelbrotlaw. The first result that follows is a special case of Theorem 3.1, and it gives usEdmundson-Lah-Ribarič type inequality for the generalized f -divergence of theZipf–Mandelbrot law. Corollary 4.1. Let p and q be Zipf-Mandelbrot laws with parameters N ∈ N , q , q ≥ and s , s > respectively, and let m p , q and M p , q be defined in (4.2).Let f : [ m p , q , M p , q ] → R be a 3-convex function. Then we have (1 − m p , q ) (cid:20) f ( M p , q ) − f ( m p , q ) M p , q − m p , q − f ′ + ( m p , q )2 (cid:21) − n X i =1 (cid:18) i + q ) s H N,q ,s − m p , q ( i + q ) s H N,q ,s (cid:19) f ′ (cid:18) H N,q ,s H N,q ,s ( i + q ) s ( i + q ) s (cid:19) ≤ M p , q − M p , q − m p , q f ( m p , q ) + 1 − m p , q M p , q − m p , q f ( M p , q ) − ˜ D f ( p , q ) (4.3) ≤ n X i =1 (cid:18) M p , q ( i + q ) s H N,q ,s − i + q ) s H N,q ,s (cid:19) f ′ (cid:18) H N,q ,s H N,q ,s ( i + q ) s ( i + q ) s (cid:19) − ( M p , q − (cid:20) f ( M p , q ) − f ( m p , q ) M p , q − m p , q − f ′− ( M p , q )2 (cid:21) . Next result follows directly from Theorem 2.3, and it gives us another Edmundson-Lah-Ribarič type inequality for the generalized f -divergence of the Zipf–Mandelbrotlaw. Corollary 4.2. Let p and q be Zipf-Mandelbrot laws with parameters N ∈ N , q , q ≥ and s , s > respectively, and let m p , q and M p , q be defined in (4.2).Let f : [ m p , q , M p , q ] → R be a 3-convex function. Then we have ( M p , q − (cid:20) f ′− ( M p , q ) − f ( M p , q ) − f ( m p , q ) M p , q − m p , q (cid:21) − f ′′− ( M p , q )2 n X i =1 ( i + q ) s H N,q ,s (cid:18) M p , q ( i + q ) s H N,q ,s − i + q ) s H N,q ,s (cid:19) ≤ M p , q − M p , q − m p , q f ( m p , q ) + 1 − m p , q M p , q − m p , q f ( M p , q ) − ˜ D f ( p , q ) (4.4) ≤ (1 − m p , q ) (cid:20) f ( M p , q ) − f ( m p , q ) M p , q − m p , q − f ′ + ( m p , q ) (cid:21) − f ′′ + ( m p , q )2 n X i =1 ( i + q ) s H N,q ,s (cid:18) i + q ) s H N,q ,s − m p , q ( i + q ) s H N,q ,s (cid:19) . Remark 4.1. By taking into consideration Example 3.1 one can see that Corol-lary 4.1 and Corollary 4.2 can easily be applied to any of the following divergences:Kullback-Leibler divergence, Hellinger divergence, Renyi divergence, harmonic di-vergence or Jeffreys divergence. NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 15 Exponential convexity Let L satisfy conditions (L1) and (L2) on a non-empty set E and let A beany positive linear functional on L with A ( ) = 1 . Let φ be a 3-convex functiondefined on an interval of real numbers I whose interior contains the interval [ m, M ] and let f ∈ L such that φ ◦ f ∈ L .Motivated by inequalities (2.1), we define following linear functionals whichrepresent the difference between the right and the left sides of the mentionedinequalities: Γ ( φ ) = M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) − A [( M − f )( f − m )] M − m (cid:18) φ ( M ) − φ ( m ) M − m − φ ′ ( m ) (cid:19) (5.1) Γ ( φ ) = A [( M − f )( f − m )] M − m (cid:18) φ ′ ( M ) − φ ( M ) − φ ( m ) M − m (cid:19) − M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) (5.2)From Theorem 2.1 it follows that functionals Γ and Γ are positive linearfunctionals under aforementioned assumptions.If function φ is in addition differentiable on h m, M i , then motivated by seriesof inequalities (2.5) and (2.9), we define following linear functionals: Γ ( φ ) = M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) − ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ ( m )2 (cid:21) − A [( f − m ) φ ′ ( f )] (5.3) Γ ( φ ) = 12 A [( M − f ) φ ′ ( f )] − ( M − A ( f )) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ ( M )2 (cid:21) − M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) . (5.4) Γ ( φ ) = M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) − ( M − A ( f )) (cid:20) φ ′ ( M ) − φ ( M ) − φ ( m ) M − m (cid:21) − φ ′′ ( M )2 A [( M − f ) ] (5.5) Γ ( φ ) = ( A ( f ) − m ) (cid:20) φ ( M ) − φ ( m ) M − m − φ ′ ( m ) (cid:21) − φ ′′ ( m )2 A [( f − m ) ] − M − A ( f ) M − m φ ( m ) + A ( f ) − mM − m φ ( M ) − A ( φ ( f )) . (5.6)Functionals Γ , Γ , Γ and Γ respectively represent the difference between theright and the left sides of the aforementioned inequalities and from Theorem 2.2and 2.3 it follows that under the assumptions from above they are positive.Now, let [ m, M ] ⊂ R be an interval such that m ≤ ≤ M . Let f be a 3-convexfunction on the interval I whose interior contains [ m, M ] and differentiable on h m, M i . Let p = ( p , ..., p n ) and p = ( q , ..., q n ) be probability distributions suchthat p i /q i ∈ [ m, M ] for every i = 1 , ..., n . Following linear functionals arise fromseries of inequalities (3.3) and (3.4) : Γ ( f ) = M − M − m f ( m ) + 1 − mM − m f ( M ) − ˜ D f ( p , q ) − (1 − m ) (cid:20) f ( M ) − f ( m ) M − m − f ′ + ( m )2 (cid:21) − n X i =1 ( p i − mq i ) f ′ (cid:18) p i q i (cid:19) (5.7) Γ ( f ) = 12 n X i =1 ( M q i − p i ) f ′ (cid:18) p i q i (cid:19) − ( M − (cid:20) f ( M ) − f ( m ) M − m − f ′− ( M )2 (cid:21) − M − M − m f ( m ) + 1 − mM − m f ( M ) − ˜ D f ( p , q ) (5.8) Γ ( f ) = M − M − m f ( m ) + 1 − mM − m f ( M ) − ˜ D f ( p , q ) − ( M − (cid:20) f ′− ( M ) − f ( M ) − f ( m ) M − m (cid:21) − f ′′− ( M )2 n X i =1 ( M q i − p i ) q i (5.9) Γ ( f ) = (1 − m ) (cid:20) f ( M ) − f ( m ) M − m − f ′ + ( m ) (cid:21) − f ′′ + ( m )2 n X i =1 ( p i − mq i ) q i − M − M − m f ( m ) + 1 − mM − m f ( M ) − ˜ D f ( p , q ) (5.10)where ˜ D f ( p , q ) is the generalized f-divergence of the probability distributions p and q defined in (3.2). From Theorem 3.1 and 3.2 it immediately follows that Γ i , i = 7 , ..., are positive linear functionals on the class of 3-convex functions onthe interval I whose interior contains [ m, M ] that are differentiable on h m, M i .First we will give some definitions and basic results regarding the exponentialconvexity that we need in the rest of this section. For the rest of this section I will denote an interval of real numbers. NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 17 Definition 5.1. A function f : I → R is said to be n -exponentially convex in theJensen sense on I if n X i,j =1 ξ i ξ j f (cid:18) t i + t j (cid:19) ≥ holds for all choices of ξ i ∈ R and every t i ∈ I , i = 1 , ..., n .A function f : I → R is n -exponentially convex if it is n -exponentially convex inthe Jensen sense and continuous on I . Remark 5.1. It is clear from the definition that 1-exponentially convex functionsin the Jensen sense are in fact nonnegative functions. Also, n -exponentially con-vex functions in the Jensen sense are k -exponentially convex in the Jensen sensefor every k ∈ N , k ≤ n . Definition 5.2. A function f : I → R is exponentially convex in the Jensen senseon I if it is n -exponentially convex in the Jensen sense for all n ∈ N .A function f : I → R is exponentially convex if it is exponentially convex in theJensen sense and continuous on I . Remark 5.2. It is known that f : I → R + is log -convex in the Jensen sense, i.e. f (cid:16) t + t (cid:17) ≤ f ( t ) f ( t ) for all t , t ∈ I (5.11) if and only if l f ( t ) + 2 lmf (cid:18) t + t (cid:19) + m f ( t ) ≥ holds for each l, m ∈ R and t , t ∈ I . Next results follow on the basis of the method developed by Jakšetić et. al. in[16], where it is shown how positive linear functionals can be used to constructsome new families of exponentially convex functions, so we give them with proofsomitted (see also [18], [28]). Theorem 5.1. Let Γ i , i = 1 , ..., be linear functionals defined in (5.1)-(5.10) re-spectively, with corresponding assumptions. Let J be an interval in R , and let Υ = { φ t : [ m, M ] → R | t ∈ J } be a family of differentiable functions such that for everyfour distinct points u , u , u , u ∈ [ m, M ] the mapping t [ u , u , u , u ] φ t is n -exponentially convex in the Jensen sense. Then the mapping t Γ i ( φ t ) is n -exponentially convex in the Jensen sense on J for i = 1 , ..., . If addition-ally the mapping t Γ i ( φ t ) is continuous on J for i = 1 , ..., , then it is n -exponentially convex on J . If the assumptions of Theorem 5.1 hold for all n ∈ N , then we immediately getthe following corollary. Corollary 5.1. Let Γ i , i = 1 , ..., be linear functionals defined in (5.1)-(5.10)respectively, with corresponding assumptions. Let J be an interval in R , andlet Υ = { φ t : [ m, M ] → R | t ∈ J } be a family of differentiable functions suchthat for every four distinct points u , u , u , u ∈ [ m, M ] the mapping t [ u , u , u , u ] φ t is exponentially convex in the Jensen sense. Then the mapping t Γ i ( φ t ) is exponentially convex in the Jensen sense on J for i = 1 , ..., . Ifadditionally the mapping t Γ i ( φ t ) is continuous on J for i = 1 , ..., , then itis exponentially convex on J . Corollary 5.2. Let Γ i , i = 1 , ..., be linear functionals defined in (5.1)-(5.10)respectively, with corresponding assumptions. Let J be an interval in R , andlet Υ = { φ t : [ m, M ] → R | t ∈ J } be a family of differentiable functions suchthat for every four distinct points u , u , u , u ∈ [ m, M ] the mapping t [ u , u , u , u ] φ t is -exponentially convex in the Jensen sense. Then the followingstatements hold. (i) If the mapping t Γ i ( φ t ) is continuous on J , then for r, s, t ∈ J suchthat r < s < t we have Γ i ( φ s ) t − r ≤ Γ i ( φ r ) t − s Γ i ( φ t ) s − r (5.12) for i = 1 , ..., . (ii) If the mapping t Γ i ( φ t ) is strictly positive and differentiable on J ,then for all s, t, u, v ∈ J such that s ≤ u and t ≤ v we have B s,t (Υ) ≤ B u,v (Υ) , where B s,t (Υ) = (cid:18) Γ i ( φ s )Γ i ( φ t ) (cid:19) s − t , s = t exp dds (Γ i ( φ s ))Γ i ( φ s ) ! , s = t (5.13) for i = 1 , ..., . Stolarsky-type means First we will give two mean value results, which are essential in producingcriteria under which Stolarsky-type quotients are actual means. The results beloware proven by following the steps in the proof of corresponding theorems from[16], so we omit the proof. Theorem 6.1. Let Γ i , i = 1 , ..., be linear functionals defined in (5.1)-(5.10)respectively with corresponding assumptions. Then for φ ∈ C ([ m, M ]) there exists ξ ∈ [ m, M ] such that Γ i ( φ ) = φ ′′′ ( ξ )6 Γ i ( φ ) NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 19 for i = 1 , ..., , where φ ( t ) = t . Theorem 6.2. Let Γ i , i = 1 , ..., be linear functionals defined in (5.1)-(5.10)respectively with corresponding assumptions. Let φ , φ ∈ C ([ m, M ]) . If Γ i ( φ ) =0 , then there exists ξ ∈ [ m, M ] such that Γ i ( φ )Γ i ( φ ) = φ ′′′ ( ξ ) φ ′′′ ( ξ ) for i = 1 , ..., or φ ′′′ ( ξ ) = φ ′′′ ( ξ ) = 0 . Remark 6.1. If the inverse of the function φ ′′′ φ ′′′ exists, then various kinds ofmeans can be defined by Theorem 6.2. That is, ξ = (cid:18) φ ′′′ φ ′′′ (cid:19) − (cid:18) Γ i ( φ )Γ i ( φ ) (cid:19) (6.1) for i = 1 , ..., . Let us consider the following family of functions Υ = { φ t : [ m, M ] → R | t ∈ R } , [ m, M ] ⊂ h , + ∞i , defined by φ t ( x ) = t ( t − t − x t , t = 0 , , 212 ln x, t = 0 , − x ln x, t = 1 , x ln x, t = 2 . (6.2)Since φ ′′′ t ( x ) = x t − ≥ , the functions φ t are -convex, and the function φ ( x ) = n X i,j =1 ξ i ξ j φ ti + tj ( x ) satisfies φ ′′′ ( x ) = n X i,j =1 ξ i ξ j φ ′′′ ti + tj ( x ) = (cid:16) n X i =1 ξ i e ( ti − ) ln x (cid:17) ≥ , so φ is 3-convex. Therefore we have ≤ [ u , u , u , u ] φ = n X i,j =1 ξ i ξ j [ u , u , u , u ] φ ti + tj ( x ) , so the mapping t [ u , u , u , u ] φ t is n -exponentially convex in the Jensensense. Since this holds for every n ∈ N , we see that family Υ satisfies theassumptions of Corollary 5.1. Hence, the mapping t Γ i ( φ t ) is exponentially convex in the Jensen sense. It is easy to check that that it is also continuous, sothe mappings t Γ i ( φ t ) , i = 1 , ..., , are exponentially convex.If we apply Theorem 6.2 for functions φ = φ s and φ = φ t given by (6.2), wecan conclude that there exists ξ ∈ [ m, M ] ⊂ h , + ∞i such that ξ = (cid:16) φ ′′′ s φ ′′′ t (cid:17) − (cid:16) Γ i ( φ s )Γ i ( φ t ) (cid:17) = (cid:16) Γ i ( φ s )Γ i ( φ t ) (cid:17) s − t , s = t. Therefore, B s,t (Υ ) given by (5.13) for the family of functions Υ is a mean ofthe segment [ m, M ] . The limiting cases s → t can be calculated, and are equalto: B s,t (Υ ) = (cid:18) Γ i ( φ s )Γ i ( φ t ) (cid:19) s − t , s = t, exp (cid:18) i ( φ s φ )Γ i ( φ ) − s − s + 2 s ( s − s − (cid:19) , s = t = 0 , , , exp (cid:18) Γ i ( φ )Γ i ( φ ) + 32 (cid:19) , s = t = 0 , exp (cid:18) Γ i ( φ φ )Γ i ( φ ) (cid:19) , s = t = 1 , exp (cid:18) Γ i ( φ φ )Γ i ( φ ) − (cid:19) , s = t = 2 . for i = 1 , ..., . From Corollary 5.2(ii) it follows that the means B s,t (Υ ) aremonotone in parameters s and t .Now consider a family of functions Υ = { ϕ t : [ m, M ] → R | t ∈ R } , [ m, M ] ⊂ h , + ∞i , defined by ϕ t ( x ) = t e tx , t = 016 x , t = 0 By straightforward calculation we see that ϕ ′′′ t ( x ) = e tx ≥ , so it follows thatthe functions ϕ t are -convex. The function defined by ϕ ( x ) = n X i,j =1 ξ i ξ j ϕ ti + tj ( x ) satisfies ϕ ′′′ ( x ) = n X i,j =1 ξ i ξ j ϕ ′′′ ti + tj ( x ) = (cid:16) n X i =1 ξ i e ti x (cid:17) ≥ , NEQUALITIES OF THE EDMUNDSON-LAH-RIBARIČ TYPE 21 so it is also 3-convex. Consequently it holds ≤ [ u , u , u , u ] ϕ = n X i,j =1 ξ i ξ j [ u , u , u , u ] ϕ ti + tj ( x ) , so the mapping t [ u , u , u , u ] ϕ t is n -exponentially convex in the Jensensense. Because this holds for every n ∈ N , the family Υ satisfies the assumptionsof Corollary 5.1. Therefore, the mapping t Γ i ( ϕ t ) is exponentially convexin the Jensen sense. Since it is also continuous, the mappings t Γ i ( ϕ t ) , i = 1 , ..., , are exponentially convex.If we put φ = ϕ s and φ = ϕ t in Theorem 6.2, we see that there has to exist ξ ∈ [ m, M ] ⊂ h , + ∞i such that ξ = (cid:16) ϕ ′′′ s ϕ ′′′ t (cid:17) − (cid:16) Γ i ( ϕ s )Γ i ( ϕ t ) (cid:17) = 1 s − t ln (cid:18) Γ i ( ϕ s )Γ i ( ϕ t ) (cid:19) , s = t. Consequently, M s,t (Υ ) defined by M s,t (Υ ) = s − t ln (cid:18) Γ i ( φ s )Γ i ( φ t ) (cid:19) , s = t dds (Γ i ( φ s ))Γ i ( φ s ) , s = t for the family of functions Υ is a mean of the segment [ m, M ] . The limitingcases s → t can be calculated, and are equal to: M s,t (Υ ) = s − t ln (cid:18) Γ i ( φ s )Γ i ( φ t ) (cid:19) , s = t Γ i (id · φ s )Γ i ( φ s ) − s , s = t = 0 , Γ i (id · φ )4Γ i ( φ ) , s = t = 0 , for i = 1 , ..., . Notice that this is a monotonic mean (in respect to parameters s and t ). References [1] S. Abramovich, Quasi-arithmetic means and subquadracity , J. Math. Inequal., (4), (2015),1157–1168.[2] P. R. 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Faculty of Textile Technology, University of Zagreb, Prilaz baruna Fil-ipovića 28a, 10 000 Zagreb, Croatia Email address : [email protected] Catholic University of Croatia, Ilica 242, 10 000 Zagreb, Croatia Email address : [email protected] Faculty of Textile Technology, University of Zagreb, Prilaz baruna Fil-ipovića 28a, 10 000 Zagreb, Croatia Email address ::