Popoviciu type inequalities for n convex functions via extension of Montgomery identity

DOI: 0.55/uom-206-0053 An. Şt. Univ. Ovidius Constnţ Vol. 24(3),206, 6 88 Popoviciu type inequlities for nconvex functions vi extension of Montgomery identity Asif R. Khn, Josip Pečrić nd Mrjn Prljk Abstrct Extension of Montgomery s identity is used in derivtion of Popoviciutype inequlities contining sums m pif(xi), where f is n n-convex function. Integrl nlogues nd some relted results for n-convex functions t point re lso given, s well s Ostrowski-type bounds for the integrl reminders of identities ssocited with the obtined inequlities. Introduction Pečrić 7] proved the following result (see lso 0, p.262]): Proposition. The inequlity p i f(x i ) 0 () holds for ll convex functions f if nd only if the mtuples x = (x,..., x m ), p = (p,..., p m ) R m stisfy p i = 0 nd p i x i x k 0 for k {,..., m}. (2) Key Words: n-convex functions, n-convex functions t point, Montgomery identity, Čebyšev functionl, exponentil convexity. 200 Mthemtics Subject Clssifiction: Primry 26A5, 26D5, 26D20. Received: 09.09.205 Accepted: 6.02.206 6

EXTENSION OF MONTGOMERY IDENTITY 62 Since p i x i x k = 2 p i (x i x k ) + p i (x i x k ), where y + = mx(y, 0), it is esy to see tht condition (2) is equivlent to p i = 0, p i x i = 0 nd p i (x i x k ) + 0 for k {,..., m}. (3) Let A denote the liner opertor A(f) = m p if(x i ), let w(x, t) = (x t) + nd x () x (2)... x (m) be the sequence x in scending order. Notice tht A(w(, x k )) = m p i(x i x k ) +. For t x (k), x (k+) ] we hve A(w(, t)) = A(w(, x (k) )) + (x (k) t) p i, {i:x i>x (k) } so the mpping t A(w(, t)) is liner on x (k), x (k+) ]. A(w(, x (m) ) = 0, so condition (3) is equivlent to Furthermore, p i = 0, p i x i = 0 nd p i (x i t) + 0 for every t x (), x (m) ]. (4) It turns out tht condition (4) is pproprite for extension of Proposition to the integrl cse nd the more generl clss of n-convex functions (see e.g. 0]). Definition. The nth order divided difference of function f : I R t distinct points x i, x i+,..., x i+n I =, b] R for some i N is defined recursively by: x j ; f] = f (x j ), j {i,..., i + n} x i,..., x i+n ; f] = xi+,...,xi+n;f]xi,...,xi+n;f] x i+nx i. It my esily be verified tht x i,..., x i+n ; f] = n f(x i+k ) i+n j=i,j i+k (x i+k x j ). Remrk. Let us denote x i,..., x i+n ; f] by (n) f(x i ). The vlue x i,..., x i+n ; f] is independent of the order of the points x i, x i+,..., x i+n. We cn extend this definition by including the cses in which two or more points coincide by tking respective limits.

EXTENSION OF MONTGOMERY IDENTITY 63 Definition 2. A function f : I R is clled convex of order n or nconvex if for ll choices of (n + ) distinct points x i,..., x i+n we hve (n) f(x i ) 0. If nth order derivtive f (n) exists, then f is n-convex if nd only if f (n) 0. For k n 2, function f is n-convex if nd only if f (k) exists nd is (n k)-convex. The following result is due to Popoviciu, 2] (see 4, 0, 9] lso). Proposition 2. Let n 2. Inequlity () holds for ll n-convex functions f :, b] R if nd only if the mtuples x, b] m, p R m stisfy p i x k i = 0, for ll k = 0,,..., n (5) p i (x i t) n + 0, for every t, b]. (6) In fct, Popoviciu proved stronger result - it is enough to ssume tht (6) holds for every t x (), x (mn+) ] nd then, due to (5), it is utomticlly stisfied for every t, b]. The integrl nlogue (see 3, 0, 9]) is given in the next proposition. Proposition 3. Let n 2, p :, β] R nd g :, β], b]. Then, the inequlity p(x)f(g(x)) dx 0 (7) holds for ll n-convex functions f :, b] R if nd only if p(x)g(x) k dx = 0, for ll k = 0,,..., n p(x) (g(x) t) n + dx 0, for every t, b]. (8) In this pper we will prove inequlities of type () nd (7) for n-convex functions by using the following extension of Montgomery s identity vi Tylor s formul obtined in ]. Proposition 4. Let n N, f : I R be such tht f (n) is bsolutely continuous, I R n open intervl,, b I, < b. Then the following

EXTENSION OF MONTGOMERY IDENTITY 64 identity holds f (x) = b n2 n2 f (k+) () (x ) k+2 f (t) dt + k! (k + 2) b f (k+) (b) (x b) k+2 + k! (k + 2) b (n )! T n (x, s) f (n) (s) ds (9) where T n (x, s) = (xs)n n(b) + x b (x s)n, s x, (xs)n n(b) + xb b (x s)n, x < s b. (0) In cse n = the sum n2 is empty, so identity (9) reduces to the well-known Montgomery identity (see for instnce 6]) f (x) = b f (t) dt + where P (x, s) is the Peno kernel, defined by P (x, s) = s b, s x, sb b, x < s b. P (x, s) f (s) ds The outline of the pper is s follows: in Section 2 we will use the extension of Montgomery s identity (9) to obtin inequlities of type () nd (7) for n-convex functions. In Section 3 we will give relted inequlities for n-convex functions t point, generliztion of the clss of n-convex functions introduced in 9]. In Section 4 we will will give bounds for the integrl reminders of identities obtined in erlier sections by using Čebyšev type inequlities. In the lst section we will prove certin properties of liner functionls ssocited with the obtined inequlities. In prticulr, we will construct new clsses of exponentilly convex functions nd Cuchy type mens. 2 Popoviciu type identities nd inequlities vi extension of Montgomery identity Theorem. Suppose ll the ssumptions from Proposition 4 hold nd let T n be given by (0). Furthermore, let m N, x i, b] nd p i R for

EXTENSION OF MONTGOMERY IDENTITY 65 i {, 2,..., m} be such tht m p i = 0. Then n2 p i f (x i ) = b k! (k + 2) f (k+) () + n2 p i (x i ) k+2 ] k! (k + 2) f (k+) (b) p i (x i b) k+2 ( b m ) p i T n (x i, s) f (n) (s) ds () (n )! Proof. We tke extension of Montgomery identity (9) to obtin b p i f (x i ) = f (t) dt p i b ( n2 ) f (k+) () (x i ) k+2 n2 f (k+) (b) (x i b) k+2 + p i k! (k + 2) b k! (k + 2) b + p i T n (x i, s) f (n) (s) ds. (n )! By simplifying this expressions we obtin (). We my stte its integrl version s follows. Theorem 2. Let g :, β], b] nd p :, β] R be integrble functions such tht p(x)dx = 0. Let n N, I R be n open intervl,, b I, < b, T n be given by (0) nd f : I R be such tht f (n) is bsolutely continuous. Then β n2 β p (x) f(g(x)) dx = b k! (k + 2) f (k+) () p (x) (g(x) ) k+2 dx + n2 k! (k + 2) f (k+) (b) (n )! p (x) (g(x) b) k+2 dx ( ) p (x) T n (g(x), s) dx f (n) (s).ds (2) Proof. Our required result is obtined by using extension of Montgomery identity (9) in the following expression nd then using Fubini s theorem. p (x) f(g(x)) dx ]

EXTENSION OF MONTGOMERY IDENTITY 66 Now we stte inequlities derived from the obtined identities. Theorem 3. Let ll the ssumptions of Theorem hold with the dditionl condition p i T n (x i, s) 0, for ll s, b]. (3) Then, for every nconvex function f : I R the following inequlity holds p i f (x i ) b n2 n2 k! (k + 2) f (k+) () k! (k + 2) f (k+) (b) p i (x i ) k+2 ] p i (x i b) k+2 (4) If the inequlity in (3) is reversed, then (4) holds with the reversed sign of inequlity. Proof. The function f is n-convex, so f (n) 0. Using this fct nd (3) in () we esily rrive t our required result. Remrk 2. If reverse inequlity holds in (3) then reverse inequlity holds in (4). Now we stte n importnt consequence. Theorem 4. Suppose ll the ssumptions from Theorem hold. Additionlly, let j N, 2 j n nd let x = (x,..., x m ), b] m, p = (p,..., p m ) R m stisfy (5) nd (6) with n replced by j. If f is n-convex nd n j is even, then p i f (x i ) b n2 k=j2 n2 k=j2 k! (k + 2) f (k+) () k! (k + 2) f (k+) (b) Proof. Let s, b] be fixed. Notice tht p i (x i ) k+2 p i (x i b) k+2. (5) (b )T n (x, s) = L n (x) + (b )(x s) n +, (6) where (x s)n L n (x) = + (x b)(x s) n. n

EXTENSION OF MONTGOMERY IDENTITY 67 Using the Pochhmmer symbol (y) k = y(y ) (y k + ) we hve ( ) j L (j) n (x) = (n ) j (x s) nj + (x b)(n ) j (x s) nj 0 ( ) j + (n ) j (x s) nj = (n ) j (x s) nj (j )(x s) + (n j)(x b)]. (7) Therefore, (6) nd (7) for s < x b yield d j dx j T n(x, s) = b L(j) n (x) + (n ) j (x s) nj = (n ) j (x s) nj (j )(x s) + (n j)(x )] 0, (8) b while for x < s we hve nj dj () dx j T n(x, s) = () nj b L(j) n (x) = (n ) j (s x) nj (j )(s x) + (n j)(b x)] 0. b (9) From (6) it is cler tht x dj dx T j n (x, s) is continuous for j n 2. Hence, if j n 2 nd n j is even, from (8) nd (9) we cn conclude tht the function x T n (x, s) is j-convex. Moreover, the conclusion extends to the cse j = n, i. e. the mpping x T n (x, s) is n-convex, since the mpping x dn2 dx T n2 n (x, s) is 2-convex. Now, by Proposition 2, we see tht ssumption (3) is stisfied, so inequlity (4) holds. Moreover, due to ssumption (5), m p ip (x i ) = 0 for every polynomil P of degree j, so the first j 2 terms in the inner sum in (4) vnish, i. e., the right hnd side of (4) under the ssumptions of this theorem is equl to the right hnd side of (5). Corollry. Suppose ll the ssumptions from Theorem hold. Additionlly, let j N, 2 j n, let x = (x,..., x m ), b] m, p = (p,..., p m ) R m

EXTENSION OF MONTGOMERY IDENTITY 68 stisfy (5) nd (6) with n replced by j nd denote H(x) = n2 b k! (k + 2) f (k+) () (x ) k+2 k=j2 n2 k=j2 If H is j-convex on, b] nd n j is even, then p i f(x i ) 0. k! (k + 2) f (k+) (b) (x b) k+2. (20) Proof. Applying Proposition 2 we conclude tht the right hnd side of (5) is nonnegtive for the j-convex function H. Remrk 3. For exmple, since the functions x (x ) k+2 nd x () kj (x b) k+2 re j-convex on, b], the function H given by (20) is j-convex if f (k+) () 0 nd () k+j f (k+) (b) 0 for k = j 2,..., n 2. In the reminder of the section we will stte integrl versions of the previous results, the proofs of which re nlogous to the discrete cse. Theorem 5. Let ll the ssumptions of Theorem 2 hold with the dditionl condition p (x) f(g(x)) dx p (x) T n (g(x), s) dx 0, b for ll s, b]. Then, for every nconvex function f : I R the following inequlity holds β n2 n2 k! (k + 2) f (k+) () k! (k + 2) f (k+) (b) p (x) (g(x) ) k+2 dx p (x) (g(x) b) k+2 dx ]. (2) Theorem 6. Suppose ll the ssumptions from Theorem 2 hold. Additionlly, let j N, 2 j n nd let p :, β] R nd g :, β], b] stisfy (8) with n replced by j. If f is n-convex nd n j is even, then p (x) f(g(x)) dx b n2 k=j2 n2 k=j2 k! (k + 2) f (k+) () k! (k + 2) f (k+) (b) p (x) (g(x) ) k+2 dx p (x) (g(x) b) k+2 dx.

EXTENSION OF MONTGOMERY IDENTITY 69 Corollry 2. Let j, n, f, p nd g be s in Theorem 6 nd let H be given by (20). If H is j-convex nd n j is even, then p(x)f(g(x)) dx 0. 3 Relted inequlities for n-convex functions t point In this section we will give relted results for the clss of n-convex functions t point introduced in 9]. Definition 3. Let I be n intervl in R, c point in the interior of I nd n N. A function f : I R is sid to be n-convex t point c if there exists constnt K such tht the function F (x) = f(x) K (n )! xn (22) is (n )-concve on I (, c] nd (n )-convex on I c, ). A function f is sid to be n-concve t point c if the function f is n-convex t point c. A property tht explins the nme of the clss is the fct tht function is n-convex on n intervl if nd only if it is n-convex t every point of the intervl (see 2, 9]). Pečrić, Prljk nd Witkowski in 9] studied necessry nd sufficient conditions on two liner functionls A : C(, c]) R nd B : C(c, b]) R so tht the inequlity A(f) B(f) holds for every function f tht is n-convex t c. In this section we will give inequlities of this type for prticulr liner functionls relted to the inequlities obtined in the previous section. Let e i denote the monomils e i (x) = x i, i N 0. For the rest of this section, A nd B will denote the liner functionls obtined s the difference of the left nd right hnd sides of inequlity (4) pplied to the intervls, c] nd c, b] respectively. More concretely, let T n,c] of (0) on these intervls, i. e., T n,c] (x, s) = T n c,b] (x, s) = nd T c,b] n denote the equivlent (xs)n n(c) + x c (x s)n, s x, (xs)n n(c) + xc c (x s)n, x < s c, (xs)n n(bc) + xc bc (x s)n, c s x, (xs)n n(bc) + xb bc (x s)n, x < s b. (23) (24)

EXTENSION OF MONTGOMERY IDENTITY 70 Let x, c] m, p R m, y c, b] l nd q R l nd denote A(f) = B(f) = l p i f (x i ) n2 c k! (k + 2) f (k+) () n2 k! (k + 2) f (k+) (c) p i (x i ) k+2 ] p i (x i c) k+2, (25) q i f (y i ) n2 b c k! (k + 2) f (k+) (c) n2 k! (k + 2) f (k+) (b) l q i (y i c) k+2 ] l q i (y i b) k+2. (26) Notice tht, using the newly introduced functionls A nd B, identity () pplied to the intervls, c] nd c, b] cn be written s ( c m ) A(f) = p i T n,c] (x i, s) f (n) (s) ds, (27) (n )! B(f) = (n )! c ( l q i T c,b] n (y i, s) ) f (n) (s) ds. (28) Theorem 7. Let x, c] m, p R m, y c, b] l nd q R l be such tht c ( m l p i T,c] n (x i, s) p i T,c] n (x i, s) 0, for every s, c], (29) q i T c,b] n (y i, s) 0, for every s c, b], (30) ) f (n) (s) ds = c ( l q i T c,b] n (y i, s) ) f (n) (s) ds, (3) where T n,c], T n c,b], A nd B re given by (23), (24), (25) nd (26) respectively. If f :, b] R is (n + )-convex t point c, then A(f) B(f). (32) If the inequlities in (29) nd (30) re reversed, then (32) holds with the reversed sign of inequlity.

EXTENSION OF MONTGOMERY IDENTITY 7 Proof. Let F = f K n! e n be s in Definition 3, i. e., the function F is n-concve on, c] nd n-convex on c, b]. Applying Theorem 3 to F on the intervl, c] we hve 0 A(F ) = A(f) K n! A(e n) (33) nd pplying Theorem 3 to F on the intervl c, b] we hve 0 B(F ) = B(f) K n! B(e n). (34) Identities (27) nd (28) pplied to the function e n yield ( c m ) A(e n ) = p i T n,c] (x i, s) ds, (n )! ( b l ) B(e n ) = q i T n c,b] (y i, s) ds. (n )! c Therefore, ssumption (3) is equivlent to A(e n ) = B(e n ). Now, from (33) nd (34) we obtin the stted inequlity. Remrk 4. In the proof of Theorem 7 we hve, ctully, shown tht A(f) K n! A(e n) = K n! B(e n) B(f). In fct, inequlity (32) still holds if we replce ssumption (3) with the weker ssumption tht K (B(e n ) A(e n )) 0. Corollry 3. Let j, j 2, n N, j, j 2 n, let f :, b] R be (n + )- convex t point c, let m-tuples x, c] m nd p R m stisfy (5) nd (6) with n replced by j, let l-tuples y c, b] l nd q R l stisfy l l q i yi k = 0, for ll k = 0,,..., j 2 q i (y i t) j2 + 0, for every t y (), y (ln+) ] nd let (3) holds. If n j nd n j 2 re even, then Proof. See the proof of Theorem 4. A(f) B(f).

EXTENSION OF MONTGOMERY IDENTITY 72 4 Bounds for identities relted to the Popoviciu-type inequlities Let f, h :, b] R be two Lebesgue integrble functions. We consider the Čebyšev functionl T (f, h) = ( b ) ( b ) b f(x)h(x)dx f(x)dx h(x)dx. b b b (35) The symbol L p, b] ( p < ) denotes the spce of p-power integrble functions on the intervl, b] equipped with the norm ( f p = f (t) p dt nd L, b] denotes the spce of essentilly bounded functions on, b] with the norm f = ess sup f (t). t,b] The following results cn be found in 4]. Proposition 5. Let f :, b] R be Lebesgue integrble function nd h :, b] R be n bsolutely continuous function with ( )(b )h ] 2 L, b]. Then we hve the inequlity ( T (f, h) T (f, f) 2 b ) p (x )(b x)h (x)] 2 dx) /2. (36) The constnt 2 in (36) is the best possible. Proposition 6. Let h :, b] R be monotonic nondecresing function nd let f :, b] R be n bsolutely continuous function such tht f L, b]. Then we hve the inequlity T (f, h) 2(b ) f (x )(b x)dh(x). (37) The constnt 2 in (37) is the best possible. By using the forementioned results we will obtin bounds for the integrl reminders of identities obtined in Section 2.

EXTENSION OF MONTGOMERY IDENTITY 73 For mtuples p = (p,..., p m ), x = (x,..., x m ) with x i, b], p i R (i =,..., m) such tht m i=0 p i = 0 nd the function T n defined s in (0), denote δ(s) = p i T n (x i, s), for s, b]. (38) Similrly for functions g :, β], b] nd p :, β] R such tht p(x)dx = 0, denote (s) = p (x) T n (g(x), s) dx, for s, b]. (39) Now, we re redy to stte the min results of this section. Theorem 8. Let n N, f :, b] R be such tht f (n) is n bsolutely continuous function with ( )(b )f (n+) ] 2 L, b], x i, b] nd p i R (i {,..., m}) such tht m i=0 p i = 0 nd let the functions T n, T nd δ be defined in (0), (35) nd (38) respectively. Then p i f (x i ) = + n2 b k! (k + 2) f (k+) () p i (x i ) k+2 ] n2 k! (k + 2) f (k+) (b) p i (x i b) k+2 f (n) (b) f (n) () ] (n )!(b ) δ(s)ds + R n(f;, b), (40) where the reminder R n(f;, b) stisfies the estimtion R n(f;, b) ( b (n )! 2 T (δ, δ) ) /2 (s )(b s)f (n+) (s)] 2 ds. (4) Proof. If we pply Proposition 5 for f δ nd h f (n), then we obtin b ( ) ( δ(s)f (n) b b (s)ds δ(s)ds f (s)ds) (n) b b ( /2 b T (δ, δ) (s )(b s)f (n+) (s)] ds) 2. 2 b

EXTENSION OF MONTGOMERY IDENTITY 74 Furthermore, we hve (n )! δ(s)f (n) (s)ds = f (n) (b) f (n) () ] + R n(f;, b). (n )!(b ) δ(s)ds where R n(f;, b) stisfies inequlity (4). Now from identity () we obtin (40). Here we stte the integrl version of previous theorem. Theorem 9. Let n N, f :, b] R be such tht f (n) is n bsolutely continuous function with ( )(b )f (n+) ] 2 L, b], let g :, β], b] nd p :, β] R be functions such tht p(x)dx = 0 nd let the functions T n, T nd be defined in (0), (35) nd (39) respectively. Then β n2 p (x) f(g(x)) dx = b k! (k + 2) f (k+) () n2 k! (k + 2) f (k+) (b) f (n) (b) f (n) () ] + (n )!(b ) p (x) (g(x) ) k+2 dx p (x) (g(x) b) k+2 dx (s)ds + R 2 n(f;, b), (42) where the reminder Rn(f; 2, b) stisfies the estimtion ( /2 Rn(f; 2 b b, b) T (, ) (s )(b s)f (n+) (s)] ds) 2. (n )! 2 (43) Proof. This results esily follows by proceeding s in the proof of previous theorem nd by replcing () by (2). By using Proposition 6 we obtin the following Grüss type inequlity. Theorem 0. Let n N, f :, b] R be such tht f (n) is n bsolutely continuous function with f (n+) 0 on, b], x i, b] nd p i R (i {,..., m}) such tht m i=0 p i = 0. Also, let the functions T nd δ be defined in (35) nd (38) respectively. Then we hve representtion (40) nd the reminder Rn(f;, b) stisfies the following estimtion Rn(f; b ], b) (n )! f (n) (b) + f (n) () 2 ]] f (n2) (b) f (n2) (). (44) ]

EXTENSION OF MONTGOMERY IDENTITY 75 Proof. If we pply Proposition 6 for f δ nd h f (n), then we obtin ( b ) ( δ(s)f (n) b b (s)ds δ(s)ds f (s)ds) (n) b b b Since 2(b ) δ (s )(b s)f (n+) (s)ds. (s )(b s)f (n+) (s)ds = (2s b)f (n) (s)ds ] ] = (b ) f (n) (b) + f (n) () 2 f (n2) (b) f (n2) (), (45) by using the identities () nd (45) we deduce (44). Next we give the integrl version of the bove theorem. Theorem. Let n N, f :, b] R be such tht f (n) is n bsolutely continuous function with f (n+) 0 on, b], let g :, β], b] nd p :, β] R be functions such tht p(x)dx = 0. Also, let the functions T nd be defined in (35) nd (39) respectively. Then we hve representtion (42) nd the reminder Rn(f; 2, b) stisfies the following estimtion Rn(f; 2 b ], b) (n )! f (n) (b) + f (n) () 2 ]] f (n2) (b) f (n2) (). (46) Now we stte some Ostrowski-type inequlities relted to the obtined identities. Theorem 2. Let ll the ssumptions of Theorem hold. Furthermore, let (q, r) be pir of conjugte exponents, tht is q, r, q + r =. Let f (n) L q, b] for some n N, n >. Then we hve p i f (x i ) b n2 k! (k + 2) f (k+) () n2 k! (k + 2) f (k+) (b) p i (x i ) k+2 ] p i (x i b) k+2 (n )! f (n) q p i T n (x i, ). (47) r

EXTENSION OF MONTGOMERY IDENTITY 76 The constnt on the right hnd side of (47) is shrp for < q nd the best possible for q =. Proof. Let us denote λ(s) = (n )! p i T n (x i, s). Now, by using identity () nd pplying Hölder s inequlity we obtin p i f (x i ) n2 b k! (k + 2) f (k+) () p i (x i ) k+2 n2 k! (k + 2) f (k+) (b) ] p i (x i b) k+2 f (n) q λ r. (48) ( /r, For the proof of the shrpness of the constnt ds) λ(s) r let us find function f for which the equlity in (48) is obtined. For < q < tke f to be such tht For q =, tke f such tht f (n) (s) = sgnλ(s) λ(s) /(q). f (n) (s) = sgnλ(s). Finlly, for q =, we prove tht λ(s)f (n) (s)ds mx λ(s) s,b] f (n) (s)ds (49) is the best possible inequlity. Function T n (x, ) for n = hs jump of t point x. But, for n 2 it is continuous, nd thus λ(s) is continuous. Suppose tht λ(s) ttins its mximum t s 0, b]. First we consider the cse λ(s 0 ) > 0. For ɛ smll enough we define f ɛ (s) by 0, s s 0, f ɛ (s) = ɛn! (s s 0) n, s 0 s s 0 + ɛ, (n)! (s s 0) n, s 0 + ɛ s b. We hve λ(s)f (n) ɛ (s)ds = s0+ɛ s 0 λ(s) ɛ ds = ɛ s0+ɛ s 0 λ(s)ds.

EXTENSION OF MONTGOMERY IDENTITY 77 Now, from inequlity (49) we hve Since ɛ s0+ɛ s 0 λ(s)ds λ(s 0 ) ɛ s0+ɛ s 0 ds = λ(s 0 ). s0+ɛ lim λ(s)ds = λ(s 0 ), ɛ 0 ɛ s 0 the sttement follows. In the cse λ(s 0 ) < 0, we define f ɛ (s) by (n)! (s s 0 ɛ) n, s s 0, f ɛ (s) = ɛn! (s s 0 ɛ) n, s 0 s s 0 + ɛ, 0, s 0 + ɛ s b nd the rest of the proof is the sme s bove. Now we give the integrl cse of the bove theorem. Theorem 3. Let ll the ssumptions of Theorem 2 hold. Furthermore, let (q, r) be pir of conjugte exponents, tht is q, r, q + r =. Let f (n) L q, b] for some n N, n >. Then we hve p (x) f(g(x)) dx n2 β b k! (k + 2) f (k+) () p (x) (g(x) ) k+2 dx n2 β k! (k + 2) f (k+) (b) p (x) (g(x) b) dx] k+2 (n )! f (n) q p(x)t n (g(x), s) dx. (50) r The constnt on the right hnd side of (50) is shrp for < q nd the best possible for q =. 5 Men Vlue Results nd Exponentil Convexity In this section we will prove some properties of liner functionls ssocited with the inequlities obtined in erlier sections. Under the ssumptions of

EXTENSION OF MONTGOMERY IDENTITY 78 Theorem 3 using (4) nd Theorem 5 using (2) we define the following functionls respectively: Λ (f) = p i f (x i ) n2 b k! (k + 2) f (k+) () n2 k! (k + 2) f (k+) (b) p i (x i ) k+2 ] p i (x i b) k+2 (A) Λ 2 (f) = b p(x)f(g(x))dx n2 β k! (k + 2) f (k+) () p (x) (g(x) ) k+2 dx n2 β k! (k + 2) f (k+) (b) p (x) (g(x) b) k+2 dx ] (A2) Now we give men vlue theorems for Λ k, k {, 2}. Here f 0 (x) = xn n!. Theorem 4. Let f C n, b] nd let Λ k : C n, b] R for k {, 2} be the liner functionls s defined in (A) nd (A2) respectively. Then there exists ξ k, b] for k {, 2} such tht Λ k (f) = f (n) (ξ k )Λ k (f 0 ), k {, 2} (5) Proof. Since f (n) is continuous on, b], so L f (n) (x) M for x, b] where L = min f (n) (x) nd M = mx f (n) (x). x,b] x,b] Therefore the function gives us F (x) = M xn n! f(x) = Mf 0(x) f(x) F (n) (x) = M f (n) (x) 0 i.e. F is nconvex function. Hence Λ k (F ) 0 nd we conclude tht for k {, 2} Λ k (f) MΛ k (f 0 ). Similrly, for k {, 2} we hve LΛ k (f 0 ) Λ k (f).

EXTENSION OF MONTGOMERY IDENTITY 79 Combining the two inequlities we get which gives us (5). LΛ k (f 0 ) Λ k (f) MΛ k (f 0 ) Theorem 5. Let f, h C n, b] nd let Λ k : C n, b] R for k {, 2} be the liner functionls s defined in (A) nd (A2) respectively. Then there exists ξ k, b] for k {, 2} such tht Λ k (f) Λ k (h) = f (n) (ξ k ) h (n) (ξ k ) ssuming tht both the denomintors re non-zero. Proof. Fix k {, 2}. Let h C n, b] be defined s ω = Λ k (h)f Λ k (f)h. Using Theorem 4 there exists ξ k such tht or which gives us the required result. 0 = Λ k (ω) = ω (n) (ξ k )Λ k (f 0 ) Λ k (h)f (n) (ξ k ) Λ k (f)h (n) (ξ k )]Λ k (f 0 ) = 0 Remrk 5. If the inverse of f (n) exists, then from the bove men vlue h (n) theorems we cn give generlized mens ( ) f (n) ( ) Λk (f) ξ k =, k {, 2}. (52) Λ k (h) h (n) 5. Logrithmiclly Convex Functions A number of importnt inequlities rise from logrithmic convexity of some functions. In the following definitions I is n intervl in R. Definition 4. A function f : I (0, ) is clled log convex in Jsense if the inequlity ( ) f 2 x + x 2 f (x ) f (x 2 ) 2 holds for ech x, x 2 I.

EXTENSION OF MONTGOMERY IDENTITY 80 Definition 5. 0, p. 7] A function f : I (0, ) is clled log convex if the inequlity f(λx + ( λ)x 2 ) f(x )] λ f(x 2 )] (λ) holds for ech x, x 2 I nd λ 0, ]. Remrk 6. A function log-convex in the Jsense is log-convex if it is continuous s well. 5.2 nexponentilly Convex Functions Bernstein 3] nd Widder 5] independently introduced n importnt sub-clss of convex functions, which is clled clss of exponentilly convex functions on given open intervl nd studied some properties of this newly defined clss. Pečrić nd Perić in 8] introduced the notion of nexponentilly convex functions which is in fct generliztion of the concept of exponentilly convex functions. In the present subsection, we discus the sme notion of nexponentil convexity by describing relted definitions nd some importnt results with some remrks from 8]. Definition 6. A function f : I R is nexponentilly convex in the Jsense if the inequlity n ( ) ti + t j u i u j f 0 2 i,j= holds for ech t i I nd u i R, i {,..., n}. Definition 7. A function f : I R is nexponentilly convex if it is nexponen-tilly convex in the Jsense nd continuous on I. Remrk 7. We cn see from the definition tht exponentilly convex functions in the Jsense re in fct nonnegtive functions. Also, nexponentilly convex functions in the Jsense re kexponentilly convex in the Jsense for every k N such tht k n. Definition 8. A function f : I R is exponentilly convex in the Jsense, if it is nexponentilly convex in the Jsense for ech n N. Remrk 8. A function f : I R is exponentilly convex if it is nexponentilly convex in the Jsense nd continuous on I.

EXTENSION OF MONTGOMERY IDENTITY 8 Proposition 7. If function f : I R is nexponentilly convex in the Jsense, then the mtrix ( )] m ti + t j f 2 is positive-semidefinite. Prticulrly ( )] m ti + t j det f 2 i,j= i,j= 0 for ech m N, m n nd t i I for i {,..., m}. Corollry 4. If function f : I R is exponentilly convex, then the mtrix ( )] m ti + t j f 2 i,j= is positive-semidefinite. Prticulrly ( )] m ti + t j det f 2 i,j= 0 for ech m N nd t i I for i {,..., m}. Corollry 5. If function f : I (0, ) is exponentilly convex, then f is log convex. Remrk 9. A function f : I (0, ) is log convex in Jsense if nd only if the inequlity ( ) u 2 t + t 2 f(t ) + 2u u 2 f + u 2 2 2f(t 2 ) 0 holds for ech t, t 2 I nd u, u 2 R. It follows tht positive function is log-convex in the Jsense if nd only if it is 2exponentilly convex in the Jsense. Also, using bsic convexity theory it follows tht positive function is log-convex if nd only if it is 2exponentilly convex. Here, we get our results concerning the nexponentil convexity nd exponentil convexity for our functionls Λ k, k {, 2} s defined in (A) nd (A2). Throughout the section I is n intervl in R.

EXTENSION OF MONTGOMERY IDENTITY 82 Theorem 6. Let D = {f t : t I} be clss of functions such tht the function t z 0, z,..., z n ; f t ] is nexponentilly convex in the Jsense on I for ny n + mutully distinct points z 0, z,..., z n, b]. Let Λ k be the liner functionls for k {, 2} s defined in (A) nd (A2). Then the following sttements re vlid: () The function t Λ k (f t ) is nexponentilly convex function in the Jsense on I. (b) If the function t Λ k (f t ) is continuous on I, then the function t Λ k (f t ) is nexponentilly convex on I. Proof. () Fix k {, 2}. Let us define the function ω for t i I, u i R, i {,..., n} s follows ω = n i,j= u i u j f t i +t j, 2 Since the function t z 0, z,..., z n ; f t ] is nexponentilly convex in the Jsense, therefore z 0, z,..., z n ; ω] = n i,j= u i u j z 0, z,..., z n ; f t i +t j ] 0 2 which implies tht ω is nconvex function on I nd therefore Λ k (ω) 0. Hence n u i u j Λ k (f t i +t j ) 0. 2 i,j= We conclude tht the function t Λ k (f t ) is n nexponentilly convex function on I in Jsense. (b) This prt esily follows from definition of nexponentilly convex function. As consequence of the bove theorem we give the following corollries. Corollry 6. Let D 2 = {f t : t I} be clss of functions such tht the function t z 0, z,..., z n ; f t ] is n exponentilly convex in the Jsense on I for ny n + mutully distinct points z 0, z,..., z n, b]. Let Λ k be the liner functionls for k {, 2} s defined in (A) nd (A2). Then the following sttements re vlid:

EXTENSION OF MONTGOMERY IDENTITY 83 () The function t Λ k (f t ) is exponentilly convex in the Jsense on I. (b) If the function t Λ k (f t ) is continuous on I, then the function t Λ k (f t ) is exponentilly convex on I. ( )] m (c) The mtrix Λ k is positive-semidefinite. Prticulrly, f t i +t j 2 i,j= ( det Λ k f t i +t j 2 )] m i,j= 0 for ech m N nd t i I where i {,..., m}. Proof. Proof follows directly from Theorem 6 by using definition of exponentil convexity nd Corollry 4. Corollry 7. Let D 3 = {f t : t I} be clss of functions such tht the function t z 0, z,..., z n ; f t ] is 2exponentilly convex in the Jsense on I for ny n + mutully distinct points z 0, z,..., z n, b]. Let Λ k be the liner functionls for k {, 2} s defined in (A) nd (A2). Then the following sttements re vlid: () If the function t Λ k (f t ) is continuous on I, then it is 2exponentilly convex on I. If the function t Λ k (f t ) is dditionlly positive, then it is lso log-convex on I. Moreover, the following Lypunov s inequlity holds for r < s < t, r, s, t I Λ k (f s )] tr Λ k (f r )] ts Λ k (f t )] sr. (53) (b) If the function t Λ k (f t ) is positive nd differentible on I, then for every s, t, u, v I such tht s u nd t v, we hve where µ s,t is defined s µ s,t (Λ k, D 3 ) = exp for f s, f t D 3. µ s,t (Λ k, D 3 ) µ u,v (Λ k, D 3 ) (54) ( Λk (f s) Λ k (f t) ) st ( d ds Λ k(f s) Λ k (f s) ), s t,, s = t. (55) Proof. () It follows directly form Theorem 6 nd Remrk 9. As the function t Λ k (f t ) is log-convex, i.e., ln Λ k (f t ) is convex we hve lnλ k (f s )] tr lnλ k (f r )] ts + lnλ k (f t )] sr, k {, 2} which gives us (53).

EXTENSION OF MONTGOMERY IDENTITY 84 (b) For convex function f, the inequlity f(s) f(t) s t f(u) f(v) u v (56) holds for ll s, t, u, v I R such tht s u, t v, s t, u v. Since Λ(f t ) is log convex by (c), setting f(t) = ln Λ(f t ) in (56) we hve ln Λ k (f s ) ln Λ k (f t ) s t ln Λ k(f u ) ln Λ k (f v ) u v (57) for s u, t v, s t, u v, which is equivlent to (55). The cses for s = t nd/or u = v re esily derived from (57) by tking respective limits. Remrk 0. The results from Theorem 6 nd Corollries 6 nd 7 still hold when ny two (ll) points z 0, z,..., z n, b] coincide for fmily of differentible (ntimes differentible) functions f t such tht the function t z 0, z,..., z n ; f t ] is nexponentilly convex, exponentilly convex nd 2expoenetilly convex in the Jsense respectively. Now, we give two importnt remrks nd one useful corollry from 5], which we will use in some exmples in the next section. Remrk. We will sy tht µ s,t (Λ k, Ω) defined with (55) is men if µ s,t (Λ k, Ω) b for s, t I nd k {, 2}, where Ω = {f t : t I} is fmily of functions nd, b] Dom(f t ). Theorems 6 give us the following corollry. Corollry 8. Let, b R nd Λ k be liner functionls for k {, 2}. Let Ω = {f t : t I} be fmily of functions in C 2, b]. If ( ) d 2 f s st dx 2 (ξ) b, d 2 f t dx 2 for ξ, b], s, t I, then µ s,t (Λ k, Ω) is men for k {, 2}. Remrk 2. In some exmples, we will get mens of this type: ( ) d 2 f s st dx 2 d 2 f t (ξ) = ξ, ξ, b], s t. dx 2

EXTENSION OF MONTGOMERY IDENTITY 85 6 Exmples with Applictions In this section, we use vrious clsses of functions Ω = {f t : t I}, where I is n intervl in R, to construct different exmples of exponentilly convex functions nd Stolrsky-type mens. Let us consider some exmples. Exmple. Let Ω = {ψ t : R 0, ) : t R} be fmily of functions defined by { e tx ψ t (x) = t, t 0, n x n n!, t = 0. Since dn dx ψ n t (x) = e tx > 0, the function ψ t (x) is nconvex on R for every t R nd t dn dx ψ n t (x) is exponentilly convex by definition. Using nlogous rguing s in the proof of Theorems 6, we hve tht t z 0, z,..., z n ; ψ t ] is exponentilly convex (nd so exponentilly convex in the Jsense). Using Corollry 6 we conclude tht t Λ k (ψ t ), k {, 2} re exponentilly convex in the Jsense. It is esy to see tht these mppings re continuous, so they re exponentilly convex. Assume tht t Λ k (ψ t ) > 0 for k {, 2}. By introducing convex functions ψ t in (52), we obtin the following mens: for k {, 2} ( ) st ln Λk (ψ s) Λ k (ψ t), s t, Λ M s,t (Λ k, Ω ) = k (id.ψ s) Λ k (ψ s) n s, s = t 0, Λ k (id.ψ 0) (n+)λ k (ψ 0), s = t = 0. where id stnds for identity function on R. Here M s,t (Λ k, Ω ) = ln(µ s,t (Λ k, Ω )), k {, 2} re in fct mens. Remrk 3. We observe here tht ξ, b] where, b R +. ( ) d n ψs st dx n d n ψ t (ln ξ) = ξ is men for dx n Exmple 2. Let Ω 2 = {ϕ t : (0, ) R : t R} be fmily of functions defined s ϕ t (x) = { (x) t t(t) (tn+), t {0,..., n }, (x) j ln(x) () nj j!(nj)!, t = j {0,..., n }. Since ϕ t (x) is nconvex function for x (0, ) nd t d2 dx ϕ 2 t (x) is exponentilly convex, so by the sme rguments given in previous exmple we conclude tht Λ k (ϕ t ), k {, 2} re exponentilly convex.

EXTENSION OF MONTGOMERY IDENTITY 86 We ssume tht Λ k (ϕ t ) > 0 for k {, 2}. For this fmily of convex functions we obtin the following mens: for k {, 2}, J = {0,,..., n } ( Λk (ϕ s) Λ k (ϕ t) ) st ( M s,t (Λ k, Ω 2 ) = exp () n (n )! Λ k(ϕ 0ϕ s) ( Λ k (ϕ s) exp () n (n )! Λ k(ϕ 0ϕ s) 2Λ k (ϕ s) + n ) kt + n,k t kt ), s t,, s = t J,, s = t J. Here M s,t (Λ k, Ω 2 ) = µ s,t (Λ k, Ω 2 ), k {, 2} re in fct mens. Remrk 4. Further, in this choice of fmily Ω 2, we hve ( ) d n ϕ s st dx n d n ϕ t (ξ) = ξ, dx n ξ, b], s t, where, b (0, ). So, using Remrk 2 we hve n importnt conclusion tht µ s,t (Λ k, Ω 2 ) is in fct men for k {, 2}. Exmple 3. Let Ω 3 = {θ t : (0, ) (0, ) : t (0, )} be fmily of functions defined by t θ t (x) = ex t. n/2 Since t dn dx θ n t (x) = e x t is exponentilly convex for x > 0, being the Lplce trnsform of nonnegtive function 5]. So, by sme rgument given in Exmple we conclude tht Λ k (θ t ), k {, 2} re exponentilly convex. We ssume tht Λ k (θ t ) > 0 for k {, 2}. For this fmily of functions we hve the following possible cses of µ s,t (Λ k, Ω 3 ): for k {, 2} M s,t (Λ k, Ω 3 ) = exp ( Λk (θ s) Λ k (θ t) ) st ( Λ k(id.θ s) 2 s Λ k (θ s) n 2s ), s t,, s = t. By (52), M s,t (Λ k, Ω 3 ) = ( s + t) ln µ s,t (Λ k, Ω 3 ), clss of mens. k {, 2} defines Exmple 4. Let Ω 4 = {φ t : (0, ) (0, ) : t (0, )} be fmily of functions defined by { t x φ t (x) = (ln t), t, n x n n, t =.

EXTENSION OF MONTGOMERY IDENTITY 87 Since dn dx n φ t (x) = t x = e xlnt > 0 for x > 0, so by sme rgument given in Exmple we conclude tht t Λ k (φ t ), k {, 2} re exponentilly convex. We ssume tht Λ k (φ t ) > 0 for k {, 2}. For this fmily of functions we hve the following possible cses of µ s,t (Λ k, Ω 4 ): for k {, 2} ( Λk (φ s) Λ k (φ t) ( M s,t (Λ k, Ω 4 ) = exp exp ) st Λ k(id.φ s) sλ k (φ n s) s ln) s Λ k (id.φ ) Λ k (φ ) ( (n+) ), s t,, s = t,, s = t =. By (52), M s,t (Λ k, Ω 4 ) = L(s, t) ln µ s,t, (Λ k, Ω 4 ), k {, 2} defines clss of mens, where L(s, t) is Logrithmic men defined s: L(s, t) = { st ln sln t, s t, s, s=t. (58) Remrk 5. Monotonicity of µ s,t (Λ k, Ω j ) follow form (54) for k {, 2}, j {, 2, 3, 4}. References ] A. Aglić Aljinović, J. Pečrić, nd A. Vukelić, On some Ostrowski type inequlities vi Montgomery identity nd Tylor s formul II, Tmkng Jour. Mth. 36 (4), (2005), 279-30. 2] I. A. Bloch, J. Pečrić nd M. Prljk, Generliztion of Levinson s inequlity, J. Mth. Inequl. 9 (205), 57 586. 3] S. N. Bernstein, Sur les fonctions bsolument monotones, Act Mth. 52 () (929), 66. 4] P. Cerone nd S. S. Drgomir, Some new Owstrowski-type bounds for the Čebyšev functionl nd pplictions, J. Mth. Inequl., 8() (204), 59-70. 5] J.Jkšetić nd J. Pečrić, Exponentil convexity method, J. Convex Anl., 20 () (203), 8 97. 6] D. S. Mitrinović, J. E. Pečrić, nd A. M. Fink, Inequlities for functions nd their Integrls nd Derivtives, Kluwer Acdemic Publishers, Dordrecht, 994. 7] J. Pečrić, On Jessens Inequlity for Convex Functions, III, J. Mth. Anl. Appl., 56 (99), 23-239.

EXTENSION OF MONTGOMERY IDENTITY 88 8] J. Pečrić nd J. Perić, Improvements of the Giccrdi nd the Petrović inequlity nd relted Stolrsky type mens, An. Univ. Criov Ser. Mt. Inform., 39 () (202), 65 75. 9] J. Pečrić, M. Prljk nd A. Witkowski, Liner opertor inequlity for n-convex functions t point, Mth. Ineq. Appl. 8 (205), 20 27. 0] J. E. Pečrić, F. Proschn nd Y. L. Tong, Convex functions, prtil orderings nd sttisticl pplictions, Acdemic Press, New York, 992. ] T. Popoviciu, Notes sur les fonctions convexes d orde superieur III, Mthemtic (Cluj) 6, (940), 74-86. 2] T. Popoviciu, Notes sur les fonctions convexes d orde superieur IV, Disqusitiones Mth., (940), 63-7. 3] T. Popoviciu, Notes sur les fonctions convexes d orde superieur IX, Bull. Mth. Soc. Roumine Sci. 43, (94), 85 4. 4] T. Popoviciu, Les fonctions convexes, Hermn nd Cie, Editeurs, Pris 944. 5] D. V. Widder, Necessry nd sufficient conditions for the representtion of function by doubly infinite Lplce integrl, Bull. Amer. Mth. Soc., 40 (4) (934), 32 326. Asif R. KHAN, Deprtment of Mthemticl Sciences, Fculty of Science, University of Krchi, University Rod, Krchi-75270, Pkistn. Emil: sifrk@uok.edu.pk Josip PEČARIĆ, Fculty of Textile Technology, University of Zgreb, Prilz brun Filipović 28A, 0000 Zgreb, Croti. Emil: pecric@hzu.hr Mrjn PRALJAK, Fculty of Food Technology nd Biotechnology, University of Zgreb, Pierottijev 6, Zgreb, Croti. Emil: mprljk@pbf.hr