International Journal of Analysis and Applications ISSN 2291-8639 Volume 14, Number 2 (2017), 180-192 http://www.etamaths.com GENERALIZED BETA-CONVEX FUNCTIONS AND INTEGRAL INEQUALITIES BANDAR BIN-MOHSIN1, MUHAMMAD UZAIR AWAN2, MUHAMMAD ASLAM NOOR1,3,∗ KHALIDA INAYAT NOOR3, SABAH IFTIKHAR3, AWAIS GUL KHAN2 Abstract. In this paper, we introduce the concept of generalized beta-convex functions. This new class of convex functions includes several new and previous known classes of convex functions as special cases. We derive some integral inequalities of Hermite-Hadamard type via generalized beta- convex functions. Some special cases are also discussed. Results proved in this paper can be viewed as significant new contributions in this dynamic field. 1. Introduction and Preliminaries Convexity theory had played a pivotal role in the development of every branch of pure and applied sciences. Closely related to this theory is inequality theory. In fact, it is known that every function is a convex function, if and if only, if satisfies an integral inequality. These type of integral inequalities are known as Hermite-Hadanard, Simpson, Trapeziodal and Newton. The integral inequalities are used to find the lower and upper bounds of natural phenomena. Due to their important applications in various branches of pure and applied science, the concept of convexity has been generalized and generalized using some interesting and novel techniques and ideas, see [1–4, 8–10, 13–15, 17–20, 23–25, 27, 30–32]. These developments played an crucial role to establish integral inequalities via various classes of convex functions and their variant forms. See [3–7, 11–20, 23–26, 28–30] and the references therein. Motivated and inspired by the research going on in these fields, we introduced and consider a new class of convex functions, which is called generalized beta-convex functions. We show that this class of generalized beta-convex functions includes several other classes of convex functions. We also derive some new integral inequalities via beta-convex functions. Several special cases are considered which cab be obtained from our main results. Our results can be viewed as a significant refinement and improvement of the of the known results. Techniques and ideas of this paper may stimulate further research. We now recall some known basic results and concepts, which are needed to obtain the main results. Definition 1.1 ( [32]). An interval I is said to be a p-convex set if Mp(x,y; t) = [tx p + (1 − t)yp] 1 p ∈ I for all x,y ∈ I,t ∈ [0, 1], where p = 2k + 1 or p = n m ,n = 2r + 1,m = 2t + 1 and k,r,t ∈ N. For p = 1, and p = −1, p-convex set reduces to convex set and harmonic convex set, respectively. Definition 1.2 ( [32]). Let I be a p-convex set. A function f : I → R is said to be p-convex function or belongs to the class PC(I), if f(Mp(x,y; t)) ≤ tf(x) + (1 − t)f(y), ∀x,y ∈ I,t ∈ [0, 1]. It is very much obvious that for p = 1 Definition 1.2 reduces to the definition for classical convex functions. Note that for p = −1, we have the definition of harmonically convex functions. Received 5th March, 2017; accepted 27th April, 2017; published 3rd July, 2017. 2010 Mathematics Subject Classification. 26D15, 26A51. Key words and phrases. convex; beta; function; Hermite-Hadamard; inequalities. c©2017 Authors retain the copyrights of their papers, and all open access articles are distributed under the terms of the Creative Commons Attribution License. 180 GENERALIZED BETA-CONVEX FUNCTIONS 181 Definition 1.3 ( [10]). A function f : I ⊂ R\{0}→ R is said to be harmonically convex function, if f ( xy (1 − t)x + ty ) ≤ tf(x) + (1 − t)f(y), ∀x,y ∈ I,t ∈ [0, 1]. Also note that for t = 1 2 in Definition 1.2, we have Jensen p-convex functions or mid p-convex functions. f(Mp(x,y; 1/2)) ≤ f(x) + f(y) 2 , ∀x,y ∈ I,t ∈ [0, 1]. We now define the concept of generalized bet-convex functions, which is the main motivation of this paper. Definition 1.4. Let I be a p-convex set. A function f : I → R is said to be a generalized beta-convex function, if f(Mp(x,y; t)) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y), ∀x,y ∈ I,t ∈ [0, 1],θ1,θ2 ∈ (0, 1]. For p = 1, we have beta-convex functions. f(tx + (1 − t)y) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y), ∀x,y ∈ R, t ∈ [0, 1],θ1,θ2 ∈ (0, 1]. For p = −1, we have harmonic beta-convex functions, which were introduced and studies by Noor et. al [21, 22]. f ( xy (1 − t)x + ty ) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y), ∀x,y ∈ R, t ∈ [0, 1],θ1,θ2 ∈ (0, 1]. We now consider some results, which are useful in obtaining our results. Lemma 1.1. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. Then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(Mp(x,y; t))dt. Proof. The proof follows from simple calculations. � Lemma 1.2 ( [18]). Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b. If f′ ∈ L [a,b], then, we have Rf (a,b; p) = f(a) + f(b) 2 − p bp −ap ∫ b a f(x) x1−p dx = bp −ap 2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t)f′([tap + (1 − t)bp] 1 p )dt. 2. Main Results In this section, we derive our main results. Theorem 2.1. Let f : I = [a,b] ⊂ R → R be a generalized beta-convex function. If f ∈ L [a,b], then 2f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]B(θ1 + 1,θ2 + 1). Proof. Let f be a generalized beta-convex function. Then f ([ ap + bp 2 ]1 p ) ≤ 1 4 [ f ( [tap + (1 − t)bp] 1 p ) + f ( [(1 − t)ap + tbp] 1 p )] . 182 MOHSIN, AWAN, NOOR, NOOR, IFTIKHAR AND KHAN Integrating both sides of above inequality with respect to t on [0, 1], we have 2f  [ap + bp 2 ]1 p   ≤ p bp −ap b∫ a f(x) x1−p dx. (2.1) Also f ( [tap + (1 − t)bp] 1 p ) ≤ tθ1 (1 − t)θ2f(x) + (1 − t)θ1tθ2f(y). Integrating both sides of above inequality with respect to t on [0, 1], we have p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]B(θ1 + 1,θ2 + 1). (2.2) On summation of inequalities (2.1) and (2.2) the proof is complete. � We now discuss a new special case of Theorem 2.1. If θ1 = θ = θ2 in Theorem 2.1, then we have following new result for Brecker type of generalized tgs-convex functions. Corollary 2.1. Let f : I = [a,b] ⊂ R → R be Brecker type of tgs-convex function. If f ∈ L [a,b], then 2f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]B(θ + 1,θ + 1). If θ1 = −θ = θ2 in Theorem 2.1, then we have following new result for Godunova-Levin-Dragomir type generalized tgs-convex functions. Corollary 2.2. Let f : I = [a,b] ⊂ R → R be Godunova-Levin-Dragomir generalized tgs-convex function. If f ∈ L [a,b], then 2f ([ ap + bp 2 ]1 p ) ≤ p bp −ap b∫ a f(x) x1−p dx ≤ [f(a) + f(b)]B(1 −θ, 1 −θ). If p = −1 in Theorem 2.1, then we have following new result for harmonic beta-convex functions. Corollary 2.3. Let f : I\{0}⊂ R → R be a harmonic beta-convex function. If f ∈ L [a,b], then, we have 2f ( 2ab a + b ) ≤ ab b−a b∫ a f(x) x2 dx ≤ [f(a) + f(b)]B(θ1 + 1,θ2 + 1). We now derive a lower bound for Hermite-Hadamard’s inequality via product of two generalized beta-convex functions. Theorem 2.2. Let f,g : I = [a,b] ⊂ R → R be two generalized beta-convex functions. If fg ∈ L [a,b], then 8f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) −B(θ1 + θ2 + 1,θ1 + θ2 + 1)M(a,b) + B(2θ1 + 1, 2θ2 + 1)N(a,b) ≤ p bp −ap 1∫ 0 f(x)g(x) x1−p dx ≤ B(2θ1 + 1, 2θ2 + 1)M(a,b) + B(θ1 + θ2 + 1,θ1 + θ2 + 1)N(a,b), GENERALIZED BETA-CONVEX FUNCTIONS 183 where M(a,b) = f(a)g(a) + f(b)g(b), (2.3) and N(a,b) = f(a)g(b) + f(b)g(a), (2.4) respectively. Proof. Since f and g are generalized beta-convex functions respectively, so f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) ≤ 1 4 [ f ( [tap + (1 − t)bp] 1 p ) + f ( [(1 − t)ap + tbp] 1 p )] × 1 4 [ g ( [tap + (1 − t)bp] 1 p ) + g ( [(1 − t)ap + tbp] 1 p )] = 1 16 [ f ( [tap + (1 − t)bp] 1 p ) g ( [tap + (1 − t)bp] 1 p ) +f ( [(1 − t)ap + tbp] 1 p ) g ( [(1 − t)ap + tbp] 1 p ) +f ( [tap + (1 − t)bp] 1 p ) g ( [(1 − t)ap + tbp] 1 p ) +f ( [(1 − t)ap + tbp] 1 p ) g ( [tap + (1 − t)bp] 1 p )] ≤ 1 16 [ f([tap + (1 − t)bp] 1 p ) g([tap + (1 − t)bp] 1 p ) +f([(1 − t)ap + tbp] 1 p ) g([(1 − t)ap + tbp] 1 p ) +[2tθ1+θ2 (1 − t)θ1+θ2 ][f(a)g(a) + f(b)g(b)] +[t2θ1 (1 − t)2θ2 + t2θ2 (1 − t)2θ1 ][f(a)g(b) + f(b)g(a)] ] . Integrating above inequality with respect to t on [0, 1], we have f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) ≤ 1 8 [ p bp −ap 1∫ 0 f(x)g(x) x1−p dx + B(θ1 + θ2 + 1,θ1 + θ2 + 1)M(a,b) + B(2θ1 + 1, 2θ2 + 1)N(a,b) ] . (2.5) Also since f and g are generalized beta-convex functions, then f ( [tap + (1 − t)bp] 1 p ) ≤ tθ1 (1 − t)θ2f(a) + (1 − t)θ1tθ2f(b), and g ( [tap + (1 − t)bp] 1 p ) ≤ tθ1 (1 − t)θ2g(a) + (1 − t)θ1tθ2g(b). Multiplying both sides of above inequality and then integrating it with respect to t on [0, 1], we have 1∫ 0 f ( [tap + (1 − t)bp] 1 p ) g ( [tap + (1 − t)bp] 1 p ) dt ≤ f(a)g(a) 1∫ 0 tθ1 (1 − t)θ2tθ1 (1 − t)θ2 dt + f(b)g(b) 1∫ 0 tθ2+θ2 (1 − t)θ1+θ1 dt + [f(a)g(b) + f(b)g(a)] 1∫ 0 tθ1 (1 − t)θ2tθ2 (1 − t)θ1 dt. 184 MOHSIN, AWAN, NOOR, NOOR, IFTIKHAR AND KHAN This implies p bp −ap b∫ a f(x)g(x) x1−p dx ≤ B(2θ1, 2θ2 + 1)M(a,b) + B(θ1 + θ2 + 1,θ1 + θ2 + 1)N(a,b). (2.6) Combining (2.5) and (2.6) completes the proof. � Next we discuss a new special case of Theorem 2.2. If θ1 = θ = θ2 in Theorem 2.2, then we have following new result for Brecker generalized tgs-convex functions. Corollary 2.4. Let f,g : I = [a,b] ⊂ R → R be two Brecker type of tgs-convex functions. If fg ∈ L [a,b], then, we have 8f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) −B(2θ + 1, 2θ + 1)[M(a,b) + N(a,b)] ≤ p bp −ap 1∫ 0 f(x)g(x) x1−p dx ≤ B(2θ1 + 1, 2θ2 + 1)M(a,b) + B(θ1 + θ2 + 1,θ1 + θ2 + 1)N(a,b), where M(a,b) and N(a,b) are given by (2.3) and (2.4) respectively. If θ1 = −θ = θ2 in Theorem 2.2, then we have following new result for Godunova-Levin-Dragomir generalized tgs-convex functions. Corollary 2.5. Let f,g : I = [a,b] ⊂ R → R be two Godunova-Levin-Dragomir generalized tgs-convex functions. If fg ∈ L [a,b], then 8f ([ ap + bp 2 ]1 p ) g ([ ap + bp 2 ]1 p ) −B(1 − 2θ, 1 − 2θ)[M(a,b) + N(a,b)] ≤ p bp −ap 1∫ 0 f(x)g(x) x1−p dx ≤ B(2θ1 + 1, 2θ2 + 1)M(a,b) + B(θ1 + θ2 + 1,θ1 + θ2 + 1)N(a,b), where M(a,b) and N(a,b) are given by (2.3) and (2.4) respectively. If p = −1 in Theorem 2.2, then we have following new result for harmonic beta-convex functions. Corollary 2.6. Let f,g : I \{0} ⊂ R → R be two harmonic beta-convex functions. If fg ∈ L [a,b], then, we have 8f ( 2ab a + b ) g ( 2ab a + b ) −B(θ1 + θ2 + 1,θ1 + θ2 + 1)M(a,b) + B(2θ1 + 1, 2θ2 + 1)N(a,b) ≤ ab b−a 1∫ 0 f(x)g(x) x2 dx ≤ B(2θ1 + 1, 2θ2 + 1)M(a,b) + B(θ1 + θ2 + 1,θ1 + θ2 + 1)N(a,b), where M(a,b) and N(a,b) are given in (2.3) and (2.4) respectively. GENERALIZED BETA-CONVEX FUNCTIONS 185 Theorem 2.3. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If f is generalized beta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1 [ k1(θ)f(a) + k2(θ)f(b) ] , where k1(θ) := B(α + θ1 + 1,β + θ2 + 1), (2.7) and k2(θ) := B(α + θ2 + 1,β + θ1 + 1), (2.8) respectively. Proof. Using Lemma 1.1 and the fact that f is generalized beta-convex function, we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(Mp(x,y; t))dt ≤ (bp −ap)α+β+1 1∫ 0 tα(1 − t)β[tθ1 (1 − t)θ2f(a) + (1 − t)θ1tθ2f(b)]dt = (bp −ap)α+β+1 [ k1(θ)f(a) + k2(θ)f(b) ] . This completes the proof. � If θ1 = θ = θ2 in Theorem 2.3, then we have Corollary 2.7. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If f is Breckner generalized tgs-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1k(θ) [ f(a) + f(b) ] , where k(θ) := B(α + θ + 1,β + θ + 1). (2.9) If θ1 = −θ = θ2 in Theorem 2.3, then we have Corollary 2.8. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If f is Godunova-Levin-Dragomir generalized tgs-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1h(θ) [ f(a) + f(b) ] , where h(θ) := B(α−θ + 1,β −θ + 1). (2.10) If p = −1 in Theorem 2.3, then we have Corollary 2.9. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If f is harmonic beta-convex function, then b∫ a ( 1 b − 1 x )α ( 1 x − 1 a )β ( f(x) x1−p ) dx ≤ ( 1 b − 1 a )α+β+1 [ k1(θ)f(a) + k2(θ)f(b) ] , 186 MOHSIN, AWAN, NOOR, NOOR, IFTIKHAR AND KHAN where k1(θ) := B(α + θ1 + 1,β + θ2 + 1), (2.11) and k2(θ) := B(α + θ2 + 1,β + θ1 + 1), (2.12) respectively. Theorem 2.4. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f| r r−1 is generalizedbeta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1B(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } B(θ1 + 1,θ2 + 1) ]r−1 r . Proof. Using Lemma 1.1, Holder’s inequality and the fact that |f| r r−1 is generalized beta-convex func- tion, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(Mp(x,y; t))dt ≤ (bp −ap)α+β+1   1∫ 0 trα(1 − t)rβdt   1 r   1∫ 0 |f(Mp(x,y; t))| r r−1 dt   r−1 r ≤ (bp −ap)α+β+1B(rα + 1,rβ + 1)   1∫ 0 { tθ1 (1 − t)θ2|f(a)| r r−1 + (1 − t)θ2tθ1|f(b)| r r−1 } dt   r−1 r ≤ (bp −ap)α+β+1B(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } B(θ1 + 1,θ2 + 1) ]r−1 r . This completes the proof. � If θ1 = θ = θ2 in Theorem 2.4, then we have Corollary 2.10. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f| r r−1 is Breckner generalized tgs-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1B(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } B(θ + 1,θ + 1) ]r−1 r . If θ1 = −θ = θ2 in Theorem 2.4, then we have Corollary 2.11. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f| r r−1 is Godunova-Levin-Dragomir type of tgs-convex function, then, we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1B(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } B(1 −θ, 1 −θ) ]r−1 r . If p = −1 in Theorem 2.4, then we have GENERALIZED BETA-CONVEX FUNCTIONS 187 Corollary 2.12. Let f : I \{0} ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f| r r−1 is harmonic beta-convex function, then, we have b∫ a ( 1 b − 1 x )α ( 1 x − 1 a )β ( f(x) x1−p ) dx ≤ ( 1 b − 1 a )α+β+1 B(rα + 1,rβ + 1) [{ |f(a)| r r−1 + |f(b)| r r−1 } B(θ1 + 1,θ2 + 1) ]r−1 r . Theorem 2.5. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f|r is beta-convex function, then, we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1 [B(α + 1,β + 1)] r−1 r [k1(θ)|f(a)|r + k2(θ)|f(b)|r] 1 r , where k1(θ) and k2(θ) are given by (2.11) and (2.12) respectively. Proof. Using Lemma 1.1, Holder’s inequality and the fact that |f|r is beta-convex function, then b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx = (bp −ap)α+β+1 1∫ 0 tα(1 − t)βf(Mp(x,y; t))dt ≤ (bp −ap)α+β+1   1∫ 0 (1 − t)αtβdt   r−1 r   1∫ 0 tα(1 − t)β |f(Mp(x,y; t))| r dt   1 r ≤ (bp −ap)α+β+1 [B(α + 1,β + 1)] r−1 r ×   1∫ 0 tα(1 − t)β [ tθ1 (1 − t)θ2|f(a)|r + (1 − t)θ1tθ2|f(b)|r ] dt   1 r = (bp −ap)α+β+1 [B(α + 1,β + 1)] r−1 r [k1(θ)|f(a)|r + k2(θ)|f(b)|r] 1 r . This completes the proof. � If θ1 = θ = θ2 in Theorem 2.5, then we have Corollary 2.13. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f|r is Breckner type of tgs-convex function, then we have b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (bp −ap)α+β+1 [B(α + 1,β + 1)] r−1 r k 1 r (t) [|f(a)|r + |f(b)|r] 1 r , where k(θ) is given by (2.9). If θ1 = −θ = θ2 in Theorem 2.5, then we have Corollary 2.14. Let f : I = [a,b] ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f|r is Godunova-Levin-Dragomir generalized tgs-convex function, then 188 MOHSIN, AWAN, NOOR, NOOR, IFTIKHAR AND KHAN b∫ a (bp −xp)α(xp −ap)β ( f(x) x1−p ) dx ≤ (b−a)α+β+1 [B(1 −α, 1 −β)] r−1 r h 1 r (t) [|f(a)|r + |f(b)|r] 1 r , where h(t) is given by (2.10). If p = −1 in Theorem 2.5, then we have Corollary 2.15. Let f : I \{0} ⊂ R → R be a continuous function such that f ∈ L [a,b]. If |f|r is harmonic beta-convex function, then, we have b∫ a ( 1 b − 1 x )α ( 1 x − 1 a )β ( f(x) x1−p ) dx ≤ ( 1 b − 1 a )α+β+1 [B(α + 1,β + 1)] r−1 r [k1(θ)|f(a)|r + k2(θ)|f(b)|r] 1 r , where k1(θ) and k2(θ) are given by (2.11) and (2.12) respectively. Now using Lemma 1.2 we derive some Hermite-Hadamard type inequalities. Theorem 2.6. Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b and f′ ∈ L [a,b]. If |f′| is beta-convex function, then |Rf (a,b; p)| ≤ bp −ap 2p [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] , where h1(θ1,θ2) := b p−1B(θ1 + 1,θ2 + 1) 2F1 (1 p − 1,θ1 + 1; θ1 + θ2 + 2; 1 − ap bp ) − 2bp−1B(θ1 + 2,θ2 + 1) 2F1 (1 p − 1,θ1 + 2; θ1 + θ2 + 3; 1 − ap bp ) , (2.13) and h2(θ1,θ2) := b p−1B(θ2 + 1,θ1 + 1) 2F1 (1 p − 1,θ2 + 1; θ1 + θ2 + 2; 1 − ap bp ) − 2bp−1B(θ2 + 2,θ1 + 1) 2F1 (1 p − 1,θ2 + 2; θ1 + θ2 + 3; 1 − ap bp ) , (2.14) respectively. GENERALIZED BETA-CONVEX FUNCTIONS 189 Proof. Using Lemma 1.2, property of the modulus and the fact that |f′| is beta-convex function, we have |Rf (a,b; p)| = ∣∣∣∣bp −ap2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t)f′([tap + (1 − t)bp] 1 p )dt ∣∣∣∣ ≤ bp −ap 2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t) [ tθ1 (1 − t)θ2|f′(a)| + (1 − t)θ1tθ2|f′(b)| ] dt = bp −ap 2p [∫ 1 0 tθ1 (1 − t)θ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(a)|dt + ∫ 1 0 (1 − t)θ1tθ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(b)|dt ] = bp −ap 2p [{ bp−1B(θ1 + 1,θ2 + 1) 2F1 (1 p − 1,θ1 + 1; θ1 + θ2 + 2; 1 − ap bp ) −2bp−1B(θ1 + 2,θ2 + 1) 2F1 (1 p − 1,θ1 + 2; θ1 + θ2 + 3; 1 − ap bp )} |f′(a)| + { bp−1B(θ2 + 1,θ1 + 1) 2F1 (1 p − 1,θ2 + 1; θ1 + θ2 + 2; 1 − ap bp ) −2bp−1B(θ2 + 2,θ1 + 1) 2F1 (1 p − 1,θ2 + 2; θ1 + θ2 + 3; 1 − ap bp )} |f′(b)| ] = bp −ap 2p [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] . This completes the proof. � We now discuss some special cases of Theorem 2.6. If θ1 = θ = θ2 in Theorem 2.6, then we have a following new result. Corollary 2.16. Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b and f′ ∈ L [a,b]. If |f′| is Breckner type of tgs-convex function, then |Rf (a,b; p)| ≤ bp −ap 2p h(θ) [|f′(a)| + |f′(b)|] , where h(θ) := bp−1B(θ + 1,θ + 1) 2F1 (1 p − 1,θ + 1; 2θ + 2; 1 − ap bp ) − 2bp−1B(θ + 2,θ + 1) 2F1 (1 p − 1,θ + 2; 2θ + 3; 1 − ap bp ) . (2.15) If θ1 = −θ = θ2 in Theorem 2.6, then we have following new result. Corollary 2.17. Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b and f′ ∈ L [a,b]. If |f′| is Godunova-Levin-Dragomir generalized tgs-convex function, then |Rf (a,b; p)| ≤ bp −ap 2p l(θ) [|f′(a)| + |f′(b)|] , where l(θ) := bp−1B(1 −θ1, 1 −θ2) 2F1 (1 p − 1, 1 −θ; 2 − 2θ; 1 − ap bp ) − 2bp−1B(2 −θ1, 1 −θ2) 2F1 (1 p − 1, 2 −θ; 3 − 2θ; 1 − ap bp ) . (2.16) If p = 1 in Theorem 2.6, then we have following new result. 190 MOHSIN, AWAN, NOOR, NOOR, IFTIKHAR AND KHAN Corollary 2.18. Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b and f′ ∈ L [a,b]. If |f′| is beta-convex function, then∣∣∣∣∣∣f(a) + f(b)2 − 1b−a 1∫ 0 f(x)dx ∣∣∣∣∣∣ ≤ b−a2 [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] , where h1(θ1,θ2) and h2(θ1,θ2) are given by (2.13) and (2.14) respectively. If p = −1 in Theorem 2.6, then we have following new result. Corollary 2.19. Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b and f′ ∈ L [a,b]. If |f′| is harmonic beta-convex function, then∣∣∣∣∣∣f(a) + f(b)2 − abb−a 1∫ 0 f(x) x2 dx ∣∣∣∣∣∣ ≤ ab(b−a)2 [h1(θ1,θ2)|f′(a)| + h2(θ1,θ2)|f′(b)|] , where h1(θ1,θ2) and h2(θ1,θ2) are given by (2.13) and (2.14) respectively. Theorem 2.7. Let f : I = [a,b] ⊂ R → R be a differentiable function on I0 (the interior of I) with a < b and f′ ∈ L [a,b]. If |f′|r is beta-convex function, then |Rf (a,b; p)| ≤ bp −ap 2p ( b1−p 2F1 (1 p − 1, 1; 2; 1 − ap bp ) − 4b1−p 2F1 (1 p − 1, 2; 3; 1 − ap bp ))1−1r × [h1(θ1,θ2)|f′(a)|rdt + h2(θ1,θ2)|f′(b)|rdt] 1 r , where h1(θ1,θ2) and h2(θ1,θ2) are given by (2.13) and (2.14) respectively. Proof. Using Lemma 1.2, property of the modulus, power mean’s inequality and the fact that |f′|r is beta-convex function, we have |Rf (a,b; p)| = ∣∣∣∣bp −ap2p ∫ 1 0 [tap + (1 − t)bp]1− 1 p (1 − 2t)f′([tap + (1 − t)bp] 1 p )dt ∣∣∣∣ ≤ bp −ap 2p (∫ 1 0 (1 − 2t)[tap + (1 − t)bp]1− 1 p dt )1−1 r ×   1∫ 0 (1 − 2t)[tap + (1 − t)bp]1− 1 p [ tθ1 (1 − t)θ2|f′(a)|r + (1 − t)θ1tθ2|f′(b)|r ] dt   1 r = bp −ap 2p (∫ 1 0 (1 − 2t)[tap + (1 − t)bp]1− 1 p dt )1−1 r × [∫ 1 0 tθ1 (1 − t)θ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(a)|rdt + ∫ 1 0 (1 − t)θ1tθ2 (1 − 2t)[tap + (1 − t)bp]1− 1 p |f′(b)|rdt ]1 r = bp −ap 2p ( b1−p 2F1 (1 p − 1, 1; 2; 1 − ap bp ) − 4b1−p 2F1 (1 p − 1, 2; 3; 1 − ap bp ))1−1r × [h1(θ1,θ2)|f′(a)|rdt + h2(θ1,θ2)|f′(b)|rdt] 1 r . This completes the proof. � GENERALIZED BETA-CONVEX FUNCTIONS 191 Acknowledgements Authors are pleased to acknowledge the ”support of Distinguished Scientist Fellowship Program (DSFP), King Saud University, Riyadh, Saudi Arabia”. References [1] W. W. 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Math. 23 (1) (2007), 130-133. 192 MOHSIN, AWAN, NOOR, NOOR, IFTIKHAR AND KHAN 1Department of Mathematics, King Saud University, Riyadh, Saudi Arabia 2Department of Mathematics, GC University, Faisalabad, Pakistan 3Department of Mathematics, COMSATS Institute of Information Technology, Islamabad, Pakistan ∗Corresponding author: noormaslam@gmail.com 1. Introduction and Preliminaries 2. Main Results Acknowledgements References