On G. Bennett s Inequality

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1 unjab University Journal of Mathematics (ISSN ) Vol.48(1)(2016) pp On G. ennett s Inequality Imran bbas aloch bdus Salam School of Mathematical Sciences, G University, Lahore, akistan. iabbasbaloch@gmail.com, iabbasbaloch@sms.edu.pk eceived: 14 December, 2015 / ccepted: 08 February, 2016 / ublished online: 18 February, 2016 bstract. The aim of this paper is to establish similar results to that of G. ennett[2].. Niculescu[4] in the context of functions which are 3-convex/concave at a point. MS (MOS) Subject lassification odes: [2010]rimary: 26D15. Secondary: 2651 Key Words:onvex function, orel measure, barycenter, 3-convex functions at a point. 1. INTODUTION vast theory developed in the study of convex functions, may be readily applied to most significant topics in real analysis economics. In past recent years, a rapid development has experienced in the theory of convex functions. This can be accredited to several causes, two of which are as follows: First, so many areas in modern analysis directly or indirectly involves application of convex functions. Second, convex functions have huge impact on the theory of inequalities several important inequalities are out-turn of the application of convex functions (see [6]). In [2], G. ennett presented some consequences of an inequality describing the behavior of convex functions with respect to a mass distribution. Later,.. Niculescu proved an abstract version of this result([4]), which is shown in the next theorem. Theorem 1.1. [4] Let I is an interval carrying a positive orel measure l ; ; are three disjoint compact subintervals of I of positive measure. Then αdl(α) = l() = l() + l() (1. 1) αdl(α) + αdl(α); (1. 2) give a necessary sufficient condition under which the inequality f(α)dl(α) f(α)dl(α) + f(α)dl(α). (1. 3) is valid for every convex function f : I. 65

2 66 Imran bbas aloch Definition 1.2. (see [5], p. 179). ny real orel measure l on an interval I such that l(i) > 0 f(α)dl(α) 0 for every nonnegative convex function f : I. I is called a Steffensen-opoviciu measure. brief description of this concept is given in the following result, independently due to T. opoviciu [6]. M. Fink [3]: Lemma 1.3. Suppose that l be a real orel measure on an interval I with l(i) > 0. Then l is a Steffensen-opoviciu measure iff the following condition of endpoints positivity, (t α)dl(α) 0 (α t)dl(α) 0 I (,t] holds for every t [a, b]. I [t, ) Theorem 1.4. (see [5], p , for details) Let l is a Steffensen- opoviciu measure on an interval I. Then the inequality f(b l ) 1 f(α)dl(α) l(i) I holds for every continuous convex function f on I, here b l = 1 l(i) the barycenter of l. I αdl(α) represents Definition 1.5. [4] ny real orel measure l on an interval I such that l(i) > 0 f(α)dl(α) 0 for every nonnegative concave function f : I. I is called a dual Steffensen-opoviciu measure. Theorem 1.6. Theorem 1.1 even works if l is a real orel measure on I,, are three disjoint subintervals of I such that the restriction of l to each of the intervals is a Steffensen-opoviciu measure the restriction of l to is a dual Steffensen- opoviciu measure. Now, we consider the inequality of G. ennett for new class of functions in following section. 2. MIN ESULTS In [1], I.. aloch, J. ecaric, M. raljak defined a new class of functions which is defined as follow: Definition 2.1. Let I be a non-degenerate interval in c an interior point of it. function f : I is called 3-convex function at point c (respectively 3-concave function at point c) if there exists a constant K such that the function F (α) = f(α) K 2 α2 is concave (resp. convex) on I (, c] convex (resp. concave) on I [c, ). property that explains the name of the class is the fact that a function is 3-convex on an interval if only if it is 3-convex at every point of the interval (see [1]).

3 On G. ennett s Inequality 67 Theorem 2.2. Let I is an interval carrying a positive orel measure l ; ; ; ; ; are six disjoint compact subintervals of I of positive measure such that l() = l() + l() ; l() = l() + l() (2. 4) αdl(α) = also c I is such that αdl(α) + αdl(α) ; βdl(β) = βdl(β) + βdl(β). (2. 5) max{right end points of interval,, } Now, if α 2 dl(t) + c min {left end points of interval,, } (2. 6) α 2 dl(α) α 2 dl(α) = β 2 dl(β) + β 2 dl(β) β 2 dl(β), (2. 7) then following inequality f(β)dl(β) + holds for every 3-convex function f : I at a point c. f(β)dl(β) f(β)dl(β) (2. 8) roof. Since f is 3-convex function at point c I, then we have a constant K such that F (α) = f(α) K 2 α2 is concave on I (, c]. Therefore, for ; ;, three disjoint compact subintervals of I [c, ) of positive measure, so by reverse of the inequality ( 1. 3 ), we have 0 = K 2 F (α)dl(α) + F (α)dl(α) f(α)dl(α) + f(α)dl(α) ( α 2 dl(α) F (α)dl(α) f(α)dl(α) α 2 dl(α) ) lso, by using the fact that F (β) = f(β) K 2 β2 is convex on I [c, ). Therefore, for ; ;, three disjoint compact subintervals of I [c, ) of positive measure, so by use of the inequality ( 1. 3 ), we have 0 F (β)dl(β) + F (β)dl(β) F (β)dl(β) = f(β)dl(β) + f(β)dl(β) f(β)dl(β) K ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ) 2

4 68 Imran bbas aloch From above, we have f(α)dl(α) + f(α)dl(α) f(α)dl(α) K ( 2 0 f(β)dl(β) + f(β)dl(β) f(β)dl(β) K ( β 2 dl(β) + 2 α 2 dl(α) α 2 dl(α) ) β 2 dl(β) β 2 dl(β) ). So, K ( α 2 dl(α) α 2 dl(α) ) 2 f(β)dl(β) + f(β)dl(β) f(β)dl(β) K ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ). 2 (2. 9) y using ( 2. 7 ), we get ( 2. 8 ). emark 2.3. From the proof of the Theorem 2.2, we have K ( α 2 dl(α) α 2 dl(α) ) (2. 10) 2 f(β)dl(β) + f(β)dl(β) f(β)dl(β) K ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ) (2. 11) 2 So under assumption ( 2. 7 ), we can get a improvement of ( 2. 8 ) as follow K ( α 2 dl(α) α 2 dl(α) ) 2 { K ( = β 2 dl(β) + β 2 dl(β) β 2 dl(β) )} 2 f(β)dl(β) + f(β)dl(β) f(β)dl(β) (2. 12)

5 On G. ennett s Inequality 69 Now, we give next result which weakens the assumption ( 2. 7 ) such that inequality ( 2. 8 ) holds under this new condition. Theorem 2.4. Suppose that I is an interval carrying a positive orel measure l ; ; ; ; ; are six disjoint compact subintervals of I of positive measure such that ( 2. 4 ) ( 2. 5 ) hold with a = max{right end points of interval,, } min{left end points of interval,, } = b (2. 13) f is 3-convex at a point c for some c [a, b]. Then if (a) f (a) 0 or (b) or (c) then ( 2. 8 ) holds. α 2 dl(α) α 2 dl(α) α 2 dl(α) +(b) 0 α 2 dl(α) β 2 dl(β) + β 2 dl(β) β 2 dl(β) β 2 dl(β) + β 2 dl(β) β 2 dl(β) (a) < 0 < +(b) f is 3 convex, roof. The idea of proof is similar to proof of Theorem 2.2. Hence, by proceeding as in the proof of Theorem 2.2. From the inequality 2. 9, we have K [( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ) 2 ( α 2 dl(α) α 2 dl(α) )] f(β)dl(β) + f(β)dl(β) f(β)dl(β) ( ) Now, due to the concavity of F on I (, c] convexity of F on I [c, ), so for every distinct points α j I (, a] β j I [b, ), j = 1, 2, 3, we have [α 1, α 2, α 3 ]f K [β 1, β 2, β 3 ]f Letting α j a β j b, we get (if exists) (a) K +(b)

6 70 Imran bbas aloch Therefore, if assumptions (a) or (b) holds, then K [( β 2 dl(β)+ β 2 dl(β) β 2 dl(β) ) ( 2 α 2 dl(α)+ α 2 dl(α) α 2 dl(α) )] is positive we conclude the result. If the assumption (c) holds, the f is left continuous, f + is right continuous, they are both non-decreasing f f +. Therefore, there exists c [a, b] such that f with associated constant K = 0 again, we can deduce the result. emark 2.5. gain from the proof of Theorem 2.4, we obtain the inequalities ( ) ( ). Now, under assumption (a), (b) or (c) of Theorem 2.4, K is positive or negative or zero respectively due to argument discussed in the proof. Therefore, we get a better improvement of ( 2. 8 ) then ( ) in this case as follow K ( α 2 dl(α) α 2 dl(α) ) 2 K ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ) 2 f(β)dl(β) + f(β)dl(β) f(β)dl(β) (2. 14) Under the assumption of Theorem 2.2 with f : I is 3-concave at point c I, the reverse of inequality ( 2. 8 ) holds. Now, we give only the statement of the theorem with weaker condition which can be proved in similar way under which the reverse of inequality ( 2. 8 ) holds for f : I is 3-concave at point c I. Theorem 2.6. Suppose that I is an interval carrying a positive orel measure l ; ; ; ; ; are six disjoint compact subintervals of I of positive measure such that ( 2. 4 ) ( 2. 5 ) hold with a = max{right end points of interval,, } min{left end points of interval,, } = b (2. 15) f : I is 3-concave at a point c for some c [a, b]. Then if (a) or (b) α 2 dl(α) (a) 0 α 2 dl(α) +(b) 0 β 2 dl(β) + β 2 dl(β) β 2 dl(β)

7 On G. ennett s Inequality 71 or (c) α 2 dl(α) α 2 dl(α) β 2 dl(β) + β 2 dl(β) β 2 dl(β) f (a) < 0 < f +(b) f is 3 concave, then reverse of ( 2. 8 ) holds. emark 2.7. Similarly as in emark 2.5, we obtain the reverse of inequalities ( ) ( ) from the proof of Theorem 2.6. Now, due the convexity of F on I (, c] concavity of F on I [c, ), so for every distinct points α j I (, a] β j I [b, ), j = 1, 2, 3., we have [ α 1, α 2, α 3 ]f K [ β 1, β 2, β 3 ]f Letting α j a β j b, we get (if exists) (a) K +(b) Now, under assumption (a) or (b) or (c) of Theorem 2.6, K is negative or positive or zero respectively due to argument discussed above. Therefore, we get a better improvement in this case as follow K ( α 2 dl(α) α 2 dl(α) ) 2 K ( β 2 dl(β) + β 2 dl(β) β 2 dl(β) ) 2 f(β)dl(β) + f(β)dl(β) f(β)dl(β) (2. 16) Theorem 2.8. Suppose that I is an interval carrying a orel measure l ; ; ; ; ; are six disjoint subintervals of I with the restriction of l to each of the intervals,, is a Steffensen-opoviciu measure the restriction of l to is a dual Steffensen-opoviciu measure such that αdl(α) = also c I is such that l() = l() + l() ; l() = l() + l() (2. 17) αdl(α)+ αdl(α) ; max{right end points of interval,, } c βdl(β) = βdl(β)+ βdl(β). (2. 18) min{left end points of interval,, } (2. 19)

8 72 Imran bbas aloch Now, if α 2 dl(α) α 2 dl(α) = β 2 dl(β) + β 2 dl(β) β 2 dl(β), (2. 20) then following inequality f(β)dl(β) + holds for every f : I 3-convex function at a point c. f(β)dl(β) f(β)dl(β) (2. 21) emark 2.9. The statement of Theorem 2.8 can be weakened under the similar setting as given in Theorem 2.4 such that the inequality ( ) holds. 3. ONLUSION In this paper, we have studied the class of 3-convex/concave functions at a point which is larger then that of 3-convex/concave functions have developed some interesting techniques for this new class. Using these techniques, we have established similar results to G. ennett inequality for this class. Methods developed results proved in this paper may stimulate further research in this field. The interested researchers are encouraged to find the particular examples of the results presented in this paper. 4. KNOWLEDGMENTS The author wishes to express his heartfelt thanks to the referees for their constructive comments helpful suggestions to improve the final version of this paper. EFEENES [1] I.. aloch, J. ecaric M. raljak, Generalization of Levinson s inequality, J. Math. Inequal. 9, No. 2 (2015) [2] G. ennett, p-free l p Inequalities, mer. Math. Monthly 117, No. 4 (2010) [3]. M. Fink, best possible Hadamard inequality, Math. Inequal. ppl. 1, (1998) [4].. Niculescu, On result of G. ennett, ull. Math. Soc. Sci. Math. oumanie Tome 54, (102) No. 3 (2011) [5].. Niculescu L. E. ersson, onvex Functions their pplications, ontemporary pproach, MS ooks in Mathematics, Vol. 23, Springer-Verlag, New York, [6] J. ecaric, F. roschan Y. L. Tong, onvex Functions, artial Orderings Statistical pplication, cadmec ress, New York [7] T. opoviciu, Notes sur les fonctions convexes d ordre superieur (IX), ull. Math. Soc. oum. Sci. 43, (1941)