Generl Mthemtics Vol. 6, No. (28), 7 97 Ostrowski Grüss Čebyšev type inequlities for functions whose modulus of second derivtives re convex Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Abstrct In this pper, we estblish Ostrowski Grüss Čebyšev type inequlities involving functions whose derivtives re bounded nd whose modulus of second derivtives re convex. 2 Mthemtics Subject Clssifiction: 26D5, 65D Key words nd phrses: Ostrowski Grüss-Čebyšev inequlities, modulus of second derivtive, convex function. Introduction In 98, A. M. Ostrowski proved the following clssicl inequlity 7]: Theorem. Let f :,b] R be continous on,b] nd diffrentible on (,b), whose first derivtive f : (,b) R is bounded on (,b), i.e., f (x) M <. Then, () f (x) f (t)dt b 4 for ll x,b], where M is constnt. ( x b 2 (b ) 2 Received 6 July, 27 Accepted for publiction (in revised form) 5 Februry, 28 7 ) 2 ] (b )M,
74 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn For two bsolutely continuous functions f,g :,b] R, consider the functionl (2) T (f,g) b b provided, the involved integrls exist. f (x)g (x)dx f (x)dx b g (x) dx, In 882, P. L. Čebyšev proved tht 6], if f,g L,b], then (b )2 () T (f,g) f 2 g. In 94, G. Grüss showed tht 6], (4) T (f,g) (M m) (N n), 8 provided, m, M, n, nd N re rel numbers stisfying the condition < m f (x) M <, < n g (x) N <, for ll x,b]. During the pst few yers, mny reserchers hve given considerble ttention to the bove inequlities nd vrious generliztions, extensions nd vrints of these inequlities hve ppered in the literture, see 2], nd the references cited therein. Motivted by the recent results given in, ], in the present pper, we estblish some inequlities similr to those given by Ostrowski, Grüss, Čebyšev nd Pchptte, involving functions whose derivtives re bounded nd whose modulus of second derivtives re convex. The nlysis used in the proofs is elementry nd bsed on the use of integrl identities proved in 2]. 2 Sttement of results Let I be suitble intervl of the rel line R. A function f : I R is clled convex if f(λx ( λ)y) λf(x) ( λ)f(y),
Ostrowski Grüss Čebyšev type inequlities... 75 for ll x,y I nd λ, ]. A function f : I (, ) is sid to be log-convex if f (λx ( λ)y) f(x)] λ f(y)] λ, for ll x,y I nd λ, ] (see]). We need the following identity, i.e., the identity (5): f (x) b f(t)dt (x b b 2 )f (x) b (x t) 2 ( λ)f (( λ)x λt)dλ b b 2 (b ) (b x) f(t)dt (x b 2 )f (x) (x ) λ 2 f (( λ) λx)dλ λ 2 f (( λ)b λx)dλ, dt for x,b], where f : I R is n bsolutely continuous function on,b] nd λ, ]. We use the following nottion to simplify the detils of presenttion: nd S s (f,g) f(x)g(x) 2 (x b 2 ) f(x)g (x) g(x)f (x)] b f(x) g(t)dt g(x) f(t)dt, 2 (b ) T s (f,g) S s (f,g)dx, b
76 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn nd define. s the usul Lebesgue norm on L,b], i.e., h : ess sup h(t) for h L,b]. t,b] The following theorems del with Ostrowski type inequlities involving two functions. Theorem 2. Let f :, b] R be bsolutely continous on, b].. If f, g re convex on,b] nd f,g L,b], then S s (f,g) g(x) (2 f (x) f (t) ) f(x) (2 g (x) g (t) )] ] (5) b (x 2 (b ) 2 2 (b ) 2, 2 for ll x,b]. 2. If f, g re log-convex on,b], then (6) S s (f,g) 2 (b ) g(x) f(x) x t 2 f (x) ln A A (lna) 2 dt x t 2 g (x) ln B B (lnb) 2 dt, for ll x,b], where A f (t) f (x) > nd B g (t) g (x) >. Theorem. Let f :, b] R be bsolutely continuous on, b].. If f, g re convex on,x] nd x,b], then for x,b], where S s (f,g) g(x) M(x) f(x) N(x)], 4 M(x) { (x ) (b )2 f () 2 b ] } f (x), 4 9(x b 2 )2 (b ) 2 ( ) b x f (b) b
Ostrowski Grüss Čebyšev type inequlities... 77 nd (7) N(x) { (x ) (b )2 g () 2 b ] } g (x), 4 9(x b 2 )2 (b ) 2 ( ) b x g (b) b for x,b]. 2. If f, g log-convex on,x] nd x,b], then S s (f,g) g(x) H(x) f(x) L(x)], x,b], where H(x) { (x ) (b )2 A (lna ) 2 2A ln A 2A 2 4 b (lna ) f () ( ) } b x B (lnb ) 2 2B ln B 2B 2 b (lnb ) f (b), nd (x ) L(x) (b )2 A 2 (lna 2 ) 2 2A 2 ln A 2 2A 2 2 (8) 4 b (ln A 2 ) g () ( ) ] b x B 2 (lnb 2 ) 2 2B 2 ln B 2 2B 2 2 b (lnb 2 ) g (b), with A f (x) f () >, B f (x) f (b) >, A 2 g (x) g () > nd B 2 g (x) g (b) >. The Grüss type inequlities re embodied in the following theorems. Theorem 4. Let f :, b] R be bsolutely continuous on, b].
78 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn. If f, g re convex on,b] nd f, g L,b], then (9) T s (f,g) b 2 (b ) 2 g(x) (2 f (x) f (t) ) f(x) (2 g (x) g (t) )] E(x)dx, for ll x,b], where E(x) (x ) (b x). 2. If f, g re log-convex on,b], then () T s (f,g) 2 (b ) 2 f(x) g(x) x t 2 f ln AA (x) (lna) 2 dt x t 2 g (x) ln B B (lnb) 2 dt dx, for ll x,b], where A f (t) f (x) > nd B g (t) g (x) >. Theorem 5. Let f :, b] R be bsolutely continuous on, b].. If f, g re convex on,b] nd f,g L,b], then () T s (f,g) b 48 { (x ) g(x) ( f () f (x) ) b f(x) ( g () g (x) )] g(x) ( f (b) f (x) ) ( ) } b x f(x) ( g (b) g (x) )] dx. b
Ostrowski Grüss Čebyšev type inequlities... 79 2. If f, g re log-convex on,b], then T s (f,g) b { (x ) A (lna ) 2 2A ln A 2A 2 (2) 4 b (lna ) ] g(x) f () A 2 (lna 2 ) 2 2A 2 ln A 2 2A 2 2 (ln A 2 ) f(x) g () B (lnb ) 2 2B ln B 2B 2 (ln B ) g(x) f (b) B 2 (ln B 2 ) 2 2B 2 ln B 2 2B 2 2 (lnb 2 ) ] (b x ) } f(x) g (b) dx. b The next theorem contins Čebyšev type inequlities. Theorem 6. Let f :, b] R be bsolutely continuous on, b].. If f, g re convex on,b] nd f,g L,b], then T () s (f,g) 6 (b ) 2 f (x) f 2 g (x) g ]E 2 (x)dx, for ll x,b], where E(x) (x ) (b x). 2. If f, g re log-convex on,b], then T (4) s (f,g) b (b ) f (x) g (x) x t 2 ln B B (lnb) 2 x t 2 ln A A (lna) 2 dt dt dx, for ll x,b], where A f (t) f (x) > nd B g (t) g (x) >.
8 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Proofs of Theorems Proof of Theorem 2. From the hypothesis of Theorem 2, it is esy to verify tht the following identity holds: (5) f (x) b f(t)dt (x b b 2 )f (x) b (x t) 2 ( λ)f (( λ)x λt)dλ dt. b Similrly (6) g (x) b g(t)dt (x b b 2 )g (x) b (x t) 2 ( λ)g (( λ)x λt)dλ dt, b for x,b]. Multiplying both sides of (5) nd (6) by g(x) nd f(x) respectively, dding the resulting identities nd rewriting, we hve: b f(x)g(x) g(x) f(t)dt f(x) g(t)dt 2(b ) 2 (x b 2 ) f (x)g(x) f(x)g (x)] 2 (b ) g(x) (x t) 2 ( λ)f (( λ)xλt)dλ dt f(x) (x t) 2 ( λ)g (( λ)x λt)dλ dt.
Ostrowski Grüss Čebyšev type inequlities... 8 This gives S s (f,g) 2 (b ) g(x) (7) f(x) (x t) 2 (x t) 2 ( λ)f (( λ)x λt)dλ dt ( λ)g (( λ)x λt)dλ dt, where b S s (f,g) f(x)g(x) g(x) f(t)dt f(x) g(t)dt 2(b ) 2 (x b 2 ) f (x)g(x) f(x)g (x)].. Since f, g re convex on,b], from (7), we observe tht Now S s (f,g) g(x) f(x) 2 (b ) x t 2 f (x) x t 2 g (x) ( λ) 2 dλ nd S s (f,g) 2 (b ) f(x) ( λ) 2 dλ f (t) ( λ) 2 dλ g (t) λ ( λ)dλ. We thus hve: 6 g(x) λ ( λ)dλ dt λ ( λ)dλ dt. x t 2 (2 f (x) f (t) )dt x t 2 (2 g (x) g (t) )dt 2 (b ) sup (2 f (x) f (t) ) g(x) t,b]
82 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn sup (2 g (x) g (t) ) f(x) t,b] { 2 (b ) ] b x t 2 dt g(x) sup 2 f (x) f (t) ] t,b] } f(x) sup 2 g (x) g (t) ] t,b] Now (x ) (b x) hve: (b )2 S s (f,g) 2 2 (b ) 2 2 ] b (x 2 )2 (b ) 2 ] (x ) (b x). ] (x b 2 )2 (b ). We therefore g(x) 2 f x f (t) ] f(x) 2 g (x) g (t) ]] 2. Since f, g re log-convex on,x] nd x,b], we hve from (7): S s (f,g) 2 (b ) g(x) x t 2 ( λ) f (x) λ f (t) λ dλ dt f(x) x t 2 ( λ) g (x) λ g (t) λ dλ dt 2 (b ) f(x) g(x) x t 2 f (x) ( λ) f λ (t) f (x) dλ dt x t 2 g (x) ( λ) g λ (t) g (x) dλ dt 2 (b ) g(x) x t 2 f (x) ln A A (ln A) 2 dt
Ostrowski Grüss Čebyšev type inequlities... 8 f(x) where A f (t) f (x) nd B g (t) g (x). Proof of Theorem. x t 2 g (x) ln B B (lnb) 2 dt, From the hypothesis of the Theorem, the following identity 5] holds: (8) f (x) b f(t)dt (x b b 2 )f (x) (x ) λ 2 f (( λ) λx) dλ 2 (b ) (b x) λ 2 f (( λ)b λx) dλ. similrly, (9) g (x) b g(t)dt (x b b 2 )g (x) (x ) λ 2 g (( λ) λx) dλ 2 (b ) (b x) λ 2 g (( λ)b λx) dλ. Multiplying both sides of (8) nd (9) by g(x) nd f(x), dding the re-
84 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn sulting identities nd rewriting, we hve: S s (f,g) ( ) 4 g(x) x (2) λ 2 f (( λ) λx) dλ b ( ) b x λ 2 f (( λ)b λx) dλ b ( ) f(x) x λ 2 g (( λ) λx) dλ b ( ) b x λ 2 g (( λ)b λx) dλ b (b )2.. Since f, g re convex on,b], from (2) we observe tht (2) S s (f,g) g(x) M(x) f(x) N(x)], 4 where M(x) (x ) b (b x) b λ 2 f (( λ) λx) dλ λ 2 f (( λ)b λx) dλ, nd N(x) (x ) b (b x) b λ 2 g (( λ) λx) dλ λ 2 g (( λ)b λx) dλ.
Ostrowski Grüss Čebyšev type inequlities... 85 Next, using the property of functions whose modulus of second derivtives re convex, we observe tht λ 2 f (( λ) λx) dλ f () λ 2 ( λ) f (x) λ dλ 2 f () 4 f (x). Similrly, λ 2 f (( λ)b λx) dλ 2 f (b) 4 f (x), lso λ 2 g (( λ) λx) dλ 2 g () 4 g (x), nd λ 2 g (( λ)b λx) dλ 2 g (b) 4 g (x). We thus hve { (x ) ( ) (b )2 b x M(x) f () f (b) 2 b b (x ) ( ) ] } b x f (x). b b Now We, therefore hve: (x ) b ( ( ) ] b x x b ) 2 b 4 9 2 (b ) 2. (22) M(x) { (x ) (b )2 f () 2 b ] } f (x). 4 9(x b 2 )2 (b ) 2 ( ) b x f (b) b
86 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Similrly, (2) N(x) { (x ) (b )2 g () 2 b ] } g (x). 4 9(x b 2 )2 (b ) 2 ( ) b x g (b) b The inequlities (2), (22) nd (2) estblish the required inequlity. 2. Since f, g re log-convex on,b], we hve from (2) S s (f,g) ( ) (b )2 4 g(x) x λ 2 f () λ f (x) λ dλ b ( ) b x λ 2 f (b) λ f (x) λ dλ b ( ) f(x) x λ 2 g () λ g (x) λ dλ b ( ) b x λ 2 g (b) λ g (x) λ dλ b ( ) ( ) (b )2 4 g(x) x f f () λ 2 λ (x) dλ b f () ( ) ( ) b x f f (b) λ 2 λ (x) dλ b f (b) ( ) ( ) f(x) x g g () λ 2 λ (x) dλ b g () ( ) ( ) b x g g (b) λ 2 λ (x) (24) dλ b g (b).
Ostrowski Grüss Čebyšev type inequlities... 87 For ny C >, we hve: λ 2 C λ dλ C (lnc)2 2C ln C 2C 2 (ln C). Also, let A f (x) f (), B f (x) f (b), A 2 g (x) g () nd B 2 g (x) g (b). We therefore hve from (24) for x,b], where S s (f,g) g(x) H(x) f(x) L(x)], (x H(x) ) 4 (b A (lna ) 2 2A ln A 2A 2 )2 b (ln A ) f () ( ) ] b x B (lnb ) 2 2B ln B 2B 2 b (ln B ) f (b), nd (x L(x) ) 4 (b A 2 (ln A 2 ) 2 2A 2 ln A 2 2A 2 2 )2 b (lna 2 ) g () ( ) ] b x B 2 (lnb 2 ) 2 2B 2 ln B 2 2B 2 2 (25) b (lnb 2 ) g (b).
88 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn Proof of Theorem 4. From the proof of Theorem 2, we hve b (26) f(x)g(x) g(x) f(t)dt f(x) g(t)dt 2(b ) 2 (x b 2 ) f (x)g(x) f(x)g (x)] 2 (b ) g(x) (x t) 2 ( λ)f ( λ)x λt]dλ dt f(x) (x t) 2 ( λ)g ( λ)x λt]dλ dt. Integrting w.r.t x from to b, we get (27) T s (f,g) 2 (b ) 2 g(x) f(x) (x t) 2 (x t) 2 ( λ)f (( λ)xλt)dλ dt ( λ)g (( λ)xλt)dλ dt dx. Since f, g re convex on,b], we hve from (27) g(x) f(x) T s (f,g) x t 2 f (x) x t 2 g x 2 (b ) 2 ( λ) 2 dλ f (t) ( λ) 2 dλ g (t) λ ( λ)dλ dt λ ( λ)dλ dt dx
Ostrowski Grüss Čebyšev type inequlities... 89 where b 2 (b ) 2 g(x) f(x) 2 (b ) 2 f(x) 2 (b ) 2 f x t 2 (x) f (t) 6 ] g x t 2 (x) g (t) dt 6 dx f g(x) x t 2 (x) ess sup g x t 2 (x) ess sup ] g (t) 6 { g(x) 2 f (x) f (t) ] f(x) 2 g (x) g (t) ]}E(x)dx, x t 2 dt E(x) (x ) (b x). ] dt f (t) 6 dt dx 2. Since f, g re log-convex on,b], we hve from (26) T s (f,g) 2 (b ) 2 g(x) f(x) 2 (b ) 2 x t 2 x t 2 g(x) ] dt ( λ) f (x) λ f (t) λ dλ dt ( λ) g (x) λ g (t) λ dλ dt dx x t 2 f (x) ( λ) f λ (t) f (x) dλ dt
9 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn f(x) 2 (b ) 2 f(x) x t 2 g (x) g(x) where A f (t) f (x), B g (t) g (x). ( λ) g (t) g (x) λ dλ dt dx x t 2 f (x) ln A A (lna) 2 dt x t 2 g (x) ln B B (lnb) 2 dt dx, Proof of Theorem 5 From the proof of Theorem, we hve S s (f,g) ( ) 4 g(x) x λ 2 f (( λ) λx)dλ b ( ) b x λ 2 f (( λ)b λx)dλ b ( ) f(x) x λ 2 g (( λ) λx)dλ b ( ) b x λ 2 g (( λ)b λx)dλ b (b )2. Integrting the bove w.r.t.x from to b, we hve T s (f,g)
Ostrowski Grüss Čebyšev type inequlities... 9 (b ) 4 f(x) ( ) x g(x) b λ 2 g (( λ) λx) dλ λ 2 f (( λ) λx) dλ ( ) b x g(x) λ 2 f (( λ)b λx) dλ b f(x) λ 2 g (( λ)b λx) dλ dx.. Since f, g re convex on,b], we hve T s (f,g) b ( ) x g(x) λ 2 ( λ) f () λ f (x) ] dλ 4 b f(x) λ 2 ( λ) g () λ g (x) ] dλ ( ) b x g(x) b f(x) b 48 λ 2 ( λ) f (b) λ f (x) ] dλ λ 2 ( λ) g (b) λ g (x) ] dλ dx { (x ) g(x) ( f () f (x) ) b f(x) ( g () g (x) )] g(x) ( f (b) f (x) ) f(x) ( g (b) g (x) )] ( ) } b x dx. b
92 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn 2. Since f, g re log-convex on,x], x,b], we hve from (2) T s (f,g) b ( ) ( ) x f g (x) f () λ 2 λ (x) dλ 4 b f () f (x) g (b) λ 2 ( g (x) g () ( ) b x g (x) f (b) b f (x) g (b) λ 2 ( g (x) g (b) Using the results of Theorem, we hve ) λ dλ ( ) f λ 2 λ (x) dλ f (b) ) λ dλ dx. T s (f,g) b 4 { (x ) g(x) f () A (ln A ) 2 2A ln A 2A 2 b (lna ) ] f(x) g () A 2 (ln A 2 ) 2 2A 2 ln A 2 2A 2 2 (lna 2 ) ( ) b x g(x) f (b) B (lnb ) 2 2B ln B 2B 2 b (ln B ) ]} f(x) g (b) B 2 (ln B 2 ) 2 2B 2 ln B 2 2B 2 2 (lnb 2 ) dx.
Ostrowski Grüss Čebyšev type inequlities... 9 Proof of Theorem 6 From the hypothesis of Theorem 6, nd using the identities (5) nd (6), we hve: (28) f (x) f(t)dt (x b b 2 )f (x) g (x) g(t)dt (x b b 2 )g (x) b b (x t) 2 (x t) 2 ( λ)f (( λ)x λt)dλ dt ( λ)g (( λ)x λt)dλ dt. Integrting both sides of (28) w.r.t. x from to b, we hve (29) T s (f,g) (b ) (x t) 2 (x t) 2 where T s (f,g) S s (f,g)dx nd b ( λ)f (( λ)xλt)dλ dt ( λ)g (( λ)xλt)dλ dt dx, S s (f,g) f(x)g(x) (x b 2 ) f(x)g (x) g(x)f (x)]
94 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn b f(x) g(t)dt g(x) f(t)dt (x b b 2 )2 f (x)g (x) (x b 2 ) b f (x) g(t)dt b g (x) f(t)dt f(t)dt g(t)dt. b b. Since f, g re convex on,b], we hve T s (f,g) (b ) (x t) 2 (b ) (x t) 2 (x t) 2 ( g (x) (b ) { ( λ) 2 f (x) λ ( λ) f (t) dλ dt ( λ) 2 g (x) λ ( λ) g (t) dλ dt dx (x t) 2 ( f (x) ess sup t,b] ) g (t) 6 ( f (x) f (t) 6 dt dx f ) (t) 6 ) dt ( ) g (x) ess sup g (t) b 2 (x t) 2 dt t,b] 6 dx b 6 (b ) (2 f (x) f ) (2 g (x) g )E 2 (x)dx,
Ostrowski Grüss Čebyšev type inequlities... 95 where E(x) (x ) (b x). 2. Since f, g re log-convex on,b], from (29) we observe tht T s (f,g) b (b ) x t 2 f (x) ( λ) f (t) f (x) x t 2 g (x) ( λ) g λ (t) g (x) dλ dt dx b (b ) x t 2 f (x) ln A A (lna) 2 dt x t 2 g (x) ln B B (lnb) 2 dt dx, where A f (t) f (x) nd B g (t) g (x). λ dλ dt References ] N.S. Brnet, P. Cerone, S.S. Drgomir, M.R. Pinheiro, A. Sofo, Ostrowski type inequlities for functions whose modulus of derivtives re convexnd pplictions, RGMIA Res. Rep. Collec., 5 (2) (22), 29-2. 2] P. Cerone, S.S. Drgomir, Ostrowski type inequlities for functions whose derivtives stisfy certin convexity ssumptions, Demonstrtio Mth., 7 (2) (24), 299-8.
96 Nzir Ahmd Mir, Arif Rfiq nd Muhmmd Rizwn ] S.S. Drgomir, Th. M. Rssis, (Eds.), Ostrowski type Inequlities nd Applictions in Numericl Integrtion, Kluwer Acdemic Publishers, Dordrect, 22. 4] S.S. Drgomir, A. Sofo, Ostrowski type inequlities for functions whose derivtives re convex, Proceeding of the 4th Interntionl Conference on Modelling nd Simultion, November -, 22. Victo ri University, Melbourne Austrsli. RGMIA Res. Rep. Collec., 5 (Supp) (22), Art.. 5] D. S. Mitrinovic, J.E. Pecric, A.M. Fink, Inequlities Involving Functions nd Their Integrls nd Derivtives, Kluver Acdemic Publishers, Dordrecht, 99. 6] D. S. Mitrinovic, J.E. Pecric, A.M. Fink, Clssicl nd New Inequlities in Anlysis, Kluwer Acdemic Publishers, Dordrect, 99. 7] A. Ostrowski, Über die Asolutbweichung einer differencienbren Functionen von ihren Integrlmittelwert, Comment. Mth. Hel, (98), 226-227. 8] B. G. Pchptte, A note on integrl Inequlities involving two logconvex functions, Mth. Inequl. Appl., 7 (4) (24), 5-55. 9] B. G. Pchptte, A note on ZHdmrd type Integrl Inequlities involving severl log-convex functions, Tmkng J. Mth., 6 () (25), 4-47. ] B. G. Pchptte, Mthemticl Inequlities, North-Hollnd Mthemticl Librry, Vol. 67 Elsvier, 25. ] B. G. Pchptte, On Ostrowski-Gruss-Cebysev type inequlities for functions whose modulus of derivtives re convex, JIPAM, 6 (4) (25), -4.
Ostrowski Grüss Čebyšev type inequlities... 97 2] J.E. Pecric, F.Proschn, Y.L. Tng, Convex functions, prtil orderings nd sttisticl Applictions, Acdemicx Press, New Yorek, 99. N. A. Mir, A. Rfiq, M. Rizwn COMSATS Institute of Informtion Technology Deprtment of Mthemtics Plot No., Sector H-8/ Islmbd 44, Pkistn E-mil: nmir@comsts.edu.pk, rfiq@comsts.edu.pk, mrizwn@comsts.edu.pk