Abstract
In this paper, we established some new integral inequalities for different kinds of convex functions by using some classical inequalities.
Keywords: Convex functions, m !convex functions, s !convex functions, (", m) !convex functions, log !convex functions.
1. Introduction
We recall following definitions.
The functions #: [$, %] & ', is said to be convex, if we have #(*+ - (1 ! *).) / *#(+) - (1 ! *)#(.)!
for all +, . 0 [$, %] and * 02[3,1]4 We can define starshaped functions on [3, %] which satisfy the condition
#(*+) / *#(+)!
for * 0 2[3,1]. TOADER (1984) defined the concept of 5 !convexity as the following:
Definition 1. The function #: [$, %] & ' is said to be
5 !convex, where 5 0 2[3,1], if for every +, . 0 [$, %] and * 0 2[3,1], we have:
#(*+ - 5(1 ! *).) / *#(+) - 5(1 ! *)#(.)4 Denote by 67(%) the set of the 5 !convex functions on [3, %] for which #(3) / 34
Some interesting and important inequalities for 5 !convex functions can be found in our references.
Accepted Date: 24.04.2016 Corresponding author: Ahmet Ocak AKDEMİR, PhD
Ağrı İbrahim Çeçen University, Faculty of Science and Letters, Department of Mathematics, Ağrı, TURKEY
E-mail: aocakakdemir@gmail.com
HUDZIK and MALIGRANDA (1994) considered among others the class of functions which are 8 !convex in the second sense.
Definition 2. A function #: '9& ' where '9= [3, ;), is
said to be 8 !convex in the second sense if #(<+ - >.) / <?#(+) -2>?#(.)2!
for all +, . 0 [3, ;), <, > @ 32with < - > = 1 and for some fixed 8 0 (3,1]. This class of 8 !convex functions in the second sense is usually denoted by 6?A.
8 !convexity introduced by BRECKNER (1978) as a generalization of convex functions. Also, BRECKNER (1993) proved the fact that the set valued map is 8 !convex only if the associated support function is 8 !convex function.
DRAGOMIR and FITZPATRICK (1999) proved the following Hadamard type integral inequality:
Theorem 1. Suppose that #: [3, ;) & [3, ;) is an
8 !convex function in the second sense, where 8 0 (3,1] and let $, % 0 [3, ;), $ B %. If # 0 CD[3,1], then the following inequalities hold:
(1.1)
E?FD# G$ - %
E H /% ! $ I #(+) J+ /21 #($) - #(%)8 - 1 2 K
L
The constant M =?9DD is the best possible in the second inequality in (1.1). The above inequalities are sharp.
Several properties of 8 !convexity in the first sense are discussed in the paper that is written by HUDZIK and MALIGRANDA (1994). Obviously, 8 !convexity means just convexity when 8 = 1. Some new Hermite Hadamard type inequalities based on concavity and 8 !convexity established by KIRMACI et al. (2007). For related results see the papers DRAGOMIR and FITZPATRICK (1999) and KIRMACI et al (2007).
DRAGOMIR (2002) proved the following theorem.
Integral Inequalities for Some Convex Functions
M. EMİN ÖZDEMİR1 and AHMET OCAK AKDEMİR2,* 1
Uludağ University, Faculty of Education, Department of Mathematics Education, Bursa-TURKEY
2
Theorem 2. Let #: [3, ;) & ' be an 5 !convex function with 5 0 (3,1] and 3 / $ / %. If # 0 CD[3,1], then one has the inequalities (1.2) # G$ - %E H /% ! $ I 21 K #(+) - 5# N +5OE J+2 L 222222222222222/5 - 1P Q2#($) - #(%)E - 52# N$5O - # N % 5O E R
MIHEŞAN (1993) gave definition of (", m) !convexity as following;
Definition 3. The function #: [3, %) & ', % S 32is said to be
(<, 5) !convex, where (<, 5) 0 [3,1]A, if we have #(*+ - 5(1 ! *).) / *L#(+) - 5(1 ! *L)#(.) for all +, . 0 [3, %] and * 0 [3,1].
Denote by 67L(%) the class of all (<, 5) !convex functions on [3, %] for which #(3) / 3. If we choose (<, 5) = (1, 5), it can be easily seen that (<, 5) !convexity reduces to 5 !convexity and for (<, 5) = (1,1). We have ordinary convex functions on [3, %]. For the recent results based on the above definition see the papers BAKULA et al. (2006), BAKULA et al. (2008), ÖZDEMİR et al. (2010), SARIKAYA et al. (2011), SET et al. (2009), ÖZDEMİR et al. (2011).
Definition 4. (See PECARIC et al. (1992)) A function
#: T & [3, ;) is said to be UVW !convex or multiplicatively convex if UVW# is convex or equivalently if for all +, . 0 T and * 0 [3,1] one has the inequality:
(1.3)
#(*+ - (1 ! *).) / [#(+)]X[2#(.)]DFX!
We note that a UVW !convex function is convex, but the converse may not necessarily be true.
Theorem 3. (ÖZDEMİR et al. (2010)) Let #, W: [$, %] & ' be real valued non-negative convex functions and Y(+, .)(*), Z(+, .)(*):2[3,1]2& '9 are defined as the following;
Y(+, .)(*) =2DA\#(*+2 -2(1 ! *).) - #^(1 ! *)+ - *._` Z(+, .)(*) =DA[W(2*+ - (1 ! *).) - 2W((1 ! *)+ - *.)] for all * 0 [3,1], we have
(1.4) 1 % ! $ I Y G+,$ - %E H K L (*)Z G+, $ - % E H (*) J+ /P(% ! $) I #(+)W(+)1 K L J+ -a 1b [c($, %) - d($, %)] And (1.5) E (% ! $)AI I Y(+, .)(*)Z(+, .)(*) J+ J. K L K L /% ! $ I #(+)W(+)1 K L J+ -1 P [c($, %) - d($, %)] where c($, %) = #($)W($) - #(%)W(%) d($, %)2= #($)W(%) - 2#(%)W($).
The main purpose of this paper is to prove some new inequalities as above, but now for 5 !convex and 8 !convex functions by modified the mappings Y(+, .)(*) and Z(+, .)(*)4
2. Main Results
Theorem 4. Let #, W: [3, ;) & '9 be 5 !convex functions
with 5 0 (3,1] , 3 / $ B % and #, W, #W 0 CD[$, %] Y(+, .)(*), Z(+, .)(*):2[3,1]2& '9 are defined as the followings:
Y(+, .)(*) =DA\#(*+ - 5(1 ! *).) - #^(1 ! *)+ - 5*._` Z(+, .)(*) =DA\W(*+ - 5(1 ! *).) - W^(1 ! *)+ - 5*._` for all * 0 2[3,1], we have
(2.1) I Y G+,K $ - %E H L (*)Z G+, $ - % E H (*)J+ /1P I #(+)W(+) J+ K L -5P (% ! $)eA DeA-5P feDI #(+) J+ -2eAI W(+) J+ K L K L h where eD=5 - 1P iW($) - W(%)E - 5W N$5O - WN % 5O E j
eA=25 - 1P i#($) - #(%)E - 5# N$5O - # N % 5O E j and (2.2) E (% ! $)AI I Y(+, .)(*)Z(+, .)(*) J+ J. K L K L /5AE I #(+)W(+) J+ -- 1 K L 5 % ! $ I #(+) J+ I W(.) J.4 K L K L
Proof. Since # and W are 5 !convex functions, we can write Y(+, .)(*) /1E [2*#(+) - 5(1 ! *)#(.) - (1 ! *)#(+) - 5*#(.)] =DA[2#(+) - 5#(.)],! (2.3) Y G2+2,$ - %E H (*) /1E k#(+) - 5# G$ - %E Hn and analogously, if we set + = + and . =L9K
A , we can write Z(+, .)(*) /1E [*W(+) - 5(1 ! *)W(.) - (1 ! *)W(+) - 5*W(.)] =DA[2W(+) - 5W(.)], (2.4) Z G+2,$ - %E H (*) /1E kW(+) - 5W G$ - %E Hn By multiplying the inequalities (2.3) and (2.4), we get (2.5)
Y G+2,$ - %E H (*)2Z G+2,$ - %E H (*)
/1P k#(+) - 5# G$ - %E Hn kW(+) - 5W G$ - %E Hn
= f2p(q)r(q)97pNtuvw Or(q)927rNtuvw Op(q)97wpNtuvw OrNtuvw O
x h.
Integrating the above inequality with respect to + on [$, %], we obtain the following inequality:
(2.6) I Y G+2,K $ - %E H (*) L y G+2, $ - % E H (*) J+ /1P zI #(+)W(+) J+ -2K L I 5# G $ - % E H W(+) J+ K L - { 5W NLK L9KA O #(+) J+- 5A{ # NL9K A O W N L9K A O K L J+|
Using the inequalities in (1.2) and by rewriting the (2.6), the proof is completed.
Remark 1. If we choose 5 = 1, inequalities (2.1) and (2.2) reduces to (1.4) and (1.5) respectively.
Theorem 5. Let #, W: [3, ;) & ' be 8 !convex functions
in the second sense and Y(+, .)(*), Z(+, .)(*):2[3,1]2& '9 are defined as the following:
Y(+, .)(*) =DA[#(*?+ - (1 ! *)?.) - #((1 ! *)?+ - *?.)] Z(+, .)(*) =DA[W(*?+ - (1 ! *)?.) - #((1 ! *)?+ - *?.)] If #, W, #W 0 CD[$, %],2 for all * 0 [3,1], we have
I [Y($, %)(*) - 2Z($, %)(*)]J*D } / 2#($) - #(%) - W($) - W(%)8 - 1 2 and I Y($, %)(*)2Z($, %)(*)D } J* /2[c($, %) - d($, %)] kE(E8 - 1) -21 1E >(8 - 12, 8 - 1)n where ! c($, %) = #($)W($) - #(%)W(%) d($, %)2= #($)W(%) - 2#(%)W($)! and the Euler Beta function is defined by
>(+, .) =2{ *D qFD2(1 ! *)~FD
} 2J*,2 +, .2 S 3. Proof. Since # and W are 8 !convex functions in the second sense, we can write
(2.7)
Y(+, .)(*) /2X•p(q)9(DFX)•p(~)9(DFX)•p(q)9X•p(~) A
(2.8)
Z(+, .)(*) /2X•r(q)9(DFX)•r(~)9(DFX)A •r(q)9X•r(~)
If we set + = $, . = % in the above inequalities and by addition, then by integrating with respect to * over [3,1], we get: I [Y($, %)(*)Z($, %)(*)]J*D } / €2p(L)92p(K)9r(L)9r(K)A • €{ *D ?J*2 } -2{ (1 ! *)}D ?J*• =#($) - 2#(%) - W($) - W(%)8 - 1
This completes the proof of the first inequality.
For the proof of the second inequality, if we multiply the inequalities (2.7) and (2.8) for + = $, . = % and by integrating with respect to * over [3,1], we have
I [Y($, %)(*)Z($, %)(*)]J*D }
= [c($, %) - d($, %)] kE(E8 - 1) -21 1E >(8 - 12, 8 - 1)n The proof is completed.
Theorem 6. Let #, W: [3, ;) & '9 be (<, 5) !convex
fonctions with (<, 5) 0 (3,1]A , 3 / $ B % and #, W, #W 0 CD[$, %]4 Y(+, .)(*), Z(+, .)(*):2[3,1]2& '9 are defined as the followings:
Y(+, .)(*) = 21E [#(2*+2 - 52(1 ! *).2) - #(5(1 ! *)+ - 2*.)] Z(+, .)(*) = 21E [W(2*+2 - 52(1 ! *).2) - W(5(1 ! *)+2 - *.)] for all * 0 [3,1], we have
(2.9) 2I [Y($, %)(*) - Z($, %)(*)]J*D } /1E k1 - 5$1 - $ n [#($) - 2#(%) - W($) - W(%)] and (2.10) I [Y($, %)(*)Z($, %)(*)]J*D } /1P [c($, %) - 2d($, %)]($(E5 - 5$ - 1 2-2A 5E$ - 1)A- 1
where c($, %) and d($, %) as in Theorem 5.
Proof. Since #2and W are (<, 5) !convex functions, we can write
Y(+, .)(*) /Xtp(q)97(DFXt)p(~)9XA tp(~)97(DFXt)p(q) If we set + = $2 and . = %, we get
(2.11)
Y($, %)(*) /1E [^#($) - 2#(%)_(*L2- 25(1 ! *L))] and analogously, we have
(2.12)
Z($, %)(*) /1E [^W($) - 2W(%)_(*L2- 25(1 ! *L))]! By adding the inequalities (2.11) and (2.12), we get (2.13)
Y($, %)(*) - Z($, %)(*) /D
A[(2#($) - 2#(%) - W($) - W(%)2)(*L2- 25(1 ! *L))] Integrating the above inequality with respect to t on [0,1], we obtain the inequality (2.9). For the proof of the inequality (2.10), by multiplying the inequalities (2.11) and (2.12), we have
Y($, %)(*)Z($, %)(*)
/Dx[c($, %) - 2d($, %)][*AL- E52*L(1 ! *L) -25A(1 ! *L)A] By integrating the above inequality with respect to * over [3,1], we get the inequality (2.10).
Theorem 7. Let #, W: [3, ;) & '9 be logarithmically
convex functions on [3, ;) and #, W, #W 0 CD[$, %]. Y(+, .)(*), Z(+, .)(*):2[3,1] & '9 are defined as in Theorem 3, then the following inequalities hold;
(2.14)
I Y($, %)(*)Z($, %)(*)J*D }
/2DA2\2C^#($)W($), #(%)W(%)_ - C^#($)W(%), #(%)W($)_` for all * 0 [3,1], where
C^#($)W($), #(%)W(%)_ =U‚#($)W($) ! U‚2#(%)W(%)#($)W($) ! #(%)W(%) and
C^#($)W(%), #(%)W($)_ =U‚#($)W(%) ! U‚2#(%)W($)4#($)W(%) ! #(%)W($) Proof. Since #, W are UVW !convex functions on [$, %] ƒ [3, ;) , we can write
Y(+, .)(*) /1E [#X(+) - #(DFX)(.) - #(DFX)(+) - #X(.)] and
Z(+, .)(*)2/1E [WX(+) - W(DFX)(.) - W(DFX)(+) - WX(.)]4 If we set + = $ and . = %, we have
(2.15)
Y(+, .)(*) /1E [#X($) - #(DFX)(%) - #(DFX)($) - #X(%)] and
(2.16)
Z($, %)(*)2/1E [WX($) - W(DFX)(%) - W(DFX)($) - WX(%)]! By multiplying the inequalities (2.15) and (2.16), we get Y($, %)(*)Z($, %)(*)
/1P [#X($) - #(DFX)(%) - #(DFX)($) - #X(%)] × [WX($) - W(DFX)(%) - W(DFX)($) - WX(%)] By integrating the above inequality with respect to * on [3,1], we obtain the inequality (2.14).
REFERENCES
DRAGOMIR, S.S., (2002). On Some New Inequalities of Hermite-Hadamard Type For 5 !Convex Functions, Tamkang Journal of Mathematics, 33 (1).
BAKULA, M.K., PECARIC, J., RIBICIC, M., (2006). Companion inequalities to Jensen’s inequality for 5 !convex and (<, 5) !convex functions, J. Inequal. Pure Appl. Math., 7. Article 194.
ÖZDEMIR, M.E., AVCI, M. and SET, E., (2010). On some inequalities of Hermite Hadamard type via 5 !convexity, Appl. Math. Lett., 23 1065-1070.
BAKULA, M.K., ÖZDEMİR, M.E. and PECARIC, J., (2008). Hadamard type inequalities for 5 !convex and (<, 5) !convex functions, J. Inequal. Pure Appl. Math., 9. Article 96.
HUDZIK, H. and MALIGRANDA, L., (1994). Some remarks on 8 !convex functions, Aequationes Math., (48) 100-111.
DRAGOMIR S.S. and TOADER, G., (1993). Some inequalities for 5 !convex functions, Studia University Babes Bolyai, Mathematica, 38 (1), 21-28.
TOADER, G., (1984). Some generalization of the convexity, Proc. Colloq. Approx. Opt., Cluj-Napoja, 329-338.
TOADER, G., (1998). On a generalization of the convexity, Mathematica, 30 (53), 83-87.
DRAGOMIR S.S., (2002). On some new inequalities of Hermite-Hadamard type for 5 !convex functions, Tamkang Journal of Mathematics, 33 (1).
BRECKNER, W.W., (1978). Stetingkeitsaussagen fur eine Klasse verallgemeinerter konvexer funktionen in topologischen linearel Raumen, Pupl.Inst.Math., 23, 13-20.
BRECKNER, W.W., (1993). Continuity of generalized convex and generalized concave set valued functions, Rev Anal., Number Thkor. Approx., 22, 39-51.
DRAGOMIR, S.S. and FITZPATRICK, S., (1999). The Hadamard’s inequality for s-convex functions in the second sense, Demonstratio Math., 32 (4), 687-696.
KIRMACI, U.S., BAKULA, M.K., ÖZDEMİR, M.E. and PECARIC, J., (2007). Hadamard-type inequalities for 8 !convex functions, Applied Mathematics and Computation, 193, 26-35.
ÖZDEMİR, M.E., SET, E. and SARIKAYA, M.Z., (2010). Konveks Fonksiyonlar Üzerine Notlar, Atatürk University.
SARIKAYA, M.Z., SET, E. and ÖZDEMİR, M.E., (2011). Some new Hadamard’s type inequalits for co-ordinated 5 !convex and (<, 5) !convex functions, Hacettepe J. of. Math. and St., 40 219-229.
MIHEŞAN, V.G., (1993). Generalization of the convexity, Seminar of Functional Equations, Approx. and convex, Cluj-Napoca (Romania).
SET, E., SARDARI, M., ÖZDEMIR, M.E. and ROOIN, J., (2009). On generalizations of the Hadamard inequality for (<, 5) !convex functions, RGMIA Res. Rep. Coll., 12 (4), Article 4.
ÖZDEMIR, M.E., KAVURMACI, H. and SET, E., (2010). Ostrowski’s type inequalities for (<, 5) !convex functions, Kyungpook Math. J. 50, 371-378. ÖZDEMIR, M.E., AVCI, M. and KAVURMACI, H.,
(2011). Hermite-Hadamard- type inequalities via (<, 5) !convextiy, Computers and Mathematics with Applications, 61, 2614-2620.
PECARIC, J., PROSCHAN, F. and TONG, Y.L., (1992). Convex Functions, Partial Orderings and Statistical Applications, Acedemic Press, Inc.
