Wirtinger inequality using Bessel functions

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1 Mirković Advances in Difference Equations 18) 18:6 R E S E A R C H Open Access Wirtinger inequality using Bessel functions Tatjana Z Mirković 1* * Correspondence: tatjanamirkovic@visokaskolaedurs 1 College of Applied Professional Studies, Vranje, Serbia Abstract This paper presents of some new Wirtinger-type integral inequalities by using Bessel functions We establish one weighted Wirtinger inequality MSC: 6D; C1 Keywords: Wirtinger s inequality; Bessel functions 1 Introduction The Wirtinger inequality plays a very important role in the theory of approximation, the theory of Sobolev s spaces, the theory of function of several variables and functional analysis In 1916 Wirtinger established an integral inequality Theorem 11 Wirtinger inequality) Let f : R R be a continuous periodic function with period and let f L Then, if f x) dx the following inequality holds: f x) dx f x) dx, with equality if and only if f x)a cos x + b sin x, where a and b are constants Theorem 1 Let f x) be a smooth function with period Then, for all real t, [ ] f x) fx + t) dx 4 sin t f x) dx 1) Equality is attained if and only if f x)acos x + b sin x + c, where a, b, c are real constants for t equality holds always) In [1], Beesack obtained the following generalization of the Wirtinger inequality: If k > 1, f x) C 1 [, ]), f ), then f x) ) k k 1 dx k sin f k x) dx, k 1 ) k )k In [], Hall proved the following theorem: The Authors) 18 This article is distributed under the terms of the Creative Commons Attribution 4 International License which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original authors) and the source, provide a link to the Creative Commons license, and indicate if changes were made

2 Mirković Advances in Difference Equations 18) 18:6 Page of 5 Theorem 1 Suppose that k N, f x) C [, ] and f ) f )Let Hu) be an even function, increasing and strictly convex on R +, andsuchthath) H ) ; moreover, uh u) as u Then we have H f x)/f x) ) f k x) k 1)λ f k x) dx, λ λk, H), ) where λ λk, H) is determined by the equation G u) du Gu)+k 1)λ u k, Gu):uH u) Hu) 4) For each non-negative constant p, the associated Bessel equation is x d y dx + x dy dx + x p ) y 5) Since Bessel s differential equation is a second-order equation, there must be two linearly independent solutions, which are called Bessel functions These functions play important roles in many areas of applied mathematics see [, 4]) Typically the general solution is given as y a 1 J p x)+a Y p x), where a 1 and a are arbitrary constants Special functions J p x) are Bessel functions of the first kind, which are finite at x for allrealvaluesofp, andy p x) are Bessel functions of the second kind, which are singular at x The Bessel function of the first kind of order p can be determined using an infinite power series expansion as follows: J p x) + 1) k k k!ɣk+p+1) x )k+p SinceƔk +1)k!, it follows that J p x) k 1) k ) x k+p 6) k!k + p)! For integer order p, functionsj p and J p are not linearly independent, J p 1) p J p In contrast, for non-integer orders, J p and J p are linearly independent The most important Bessel functions are J x)andj 1 x) For p 1 and p 1,thisfunctions expansion as follows: J 1/ x) cos x, 7) x J 1/ x) sin x 8) x Main results Theorem 1 Let f L k on [, ], with f ) f ) Then the following inequality holds: f k x) dx 1 ) k ) ) k J k 1 k cos t cos tdt f k x) dx 9)

3 Mirković Advances in Difference Equations 18) 18:6 Page of 5 Proof Since J n z) z )n + r 1)r z )r r!n+r)!, it follows that J k cos t) + r 1)r k cos t ) J k cos t cos tdt r r cos t )r 1) r k r!) 1) r r k) r r r!) cos tdt cos r+1 tdt )r r!) Using the integration by parts formula on the integral I r+1 cosr+1 tdtand the fact that / cos tdt 1, we obtained the recurrence relation I r+1 r r+1 I r 1, which implies I r+1 r)!! r+1)!! r)!!) r+1)! r r!) r+1)! The above equality becomes which implies J k cos t ) cos tdt r k 1) r r k) r r r!) r r!) r +1)! r 1) r k )r+1 r +1)! k sin k, sin k ) k ) k k J k k cos t cos tdt) By ) it follows that f k x) dx k 1 k 1 kk 4 k k k f k x) dx ) ) k J k cos t cos tdt f k x) dx, 1 ) k ) ) k J k 1 k cos t cos tdt f k x) dx Theorem If f L k is absolutely continuous on [, ], with f ) f )then f k x) dx k k +1 Ck) f x)f k 1) x) dx, 1) where Ck): x k J1/ k x) dx x k+1 J1/ k+1x) dx Proof Starting with the right side of 1), we obtain k k +1 Ck) f x)f k 1) x) dx k k +1 xk J k 1/ xk+1 J k+1 1/ x) dx x) dx f x)f k 1) x) dx

4 Mirković Advances in Difference Equations 18) 18:6 Page 4 of 5 Ɣ 1 k xk x )k sin k xdx k +1) xk+1 x )k+1 sin k+1 xdx k sink xdx k +1 f x)f k 1) x) dx sink+1 xdx Since sinp x cos q xdx )Ɣ k+1 ) p+1 q+1 Ɣ )Ɣ ) Ɣ p+q Ɣk+1) ;Forp k +1andq,weget By integrating by parts, we obtain Ɣ k+1 Ɣ 1 ), Ɣk +1)k!, it follows k k +1 k)! k k!) k!) k+1)! f x)f k 1) x) dx,forp k and q,weget +1) sink xdx f x)f k 1) x) dx sink+1 xdx Ɣ 1 )Ɣk+1) Ɣ k+ ) k)! ) k k!, Ɣ k+ ) 1 k+ k)!) k +1) f x)f k 1) x) dx k +1 k k!) k!) ) 1 k+ ) k 1)! k f x)f k 1) x) dx k +1 kk!) If in 4)weputHu)u, Gu)u,then6)givesλ which implies f x)f k 1) x) dx 1 f k x) dx, k ) k+1)! k+1 k! 1 k k 1),so)becomes, and since k k +1 Ck) f x)f k 1) x) dx ) k+ ) k 1)! f k x) dx kk!) Since )k+ >1and k 1)! kk!) ) >1,inequality1)isestablished Theorem Let f x) be a smooth function with period Then, for all real t, [ ] f x) fx + t) dx tj t 1/ f ) x) dx 11) Equality is attained if and only if f x)a cos x+b sin x+c, where A, B, C are real constants for t equality holds always) J nt cos x) dx, forn 1, the right side of 11) be- Proof From the equation Jn t) comes t J 1 t cos x) dx f x) dx 1) n ) t cos x n+1 t dx f x) dx n!n +1)! n

5 Mirković Advances in Difference Equations 18) 18:6 Page 5 of 5 t t t t n n n n [ t t 1) n ) t n+1 cos n+1 xdx f x) dx n!n +1)! 1) n n!n +1)! t n+1 1) n n!t n+1 n +1)n!n +1)! n n n!n! n +1)! f x) dx 1) n t n+1 f x) dx n + 1)n +1)! ] 4 7! + f x) dx 1! t! + t5 5! t7 f x) dx [ t ] t!t 4t4 4!t + 6t6 6!t 8t8 4 8!t + 1t1 5 1!t f x) dx [ t ] t!t t4 4!t + t6 6!t t8 8!t + t1 1!t f x) dx t n1 t 1 1) n+1 t n f x) dx tn)! n ) 1) n tn f x) dx n)! 1 cos t) f x) dx 4sin t f x) dx Equation 1) implies thedesired inequality 11) Acknowledgements The author would like to thank the anonymous referees for their constructive comments The author states that no funding source or sponsor has participated in the realization of this work Competing interests The author declares that he has no competing interests Authors contributions The work as a whole is a contribution of the author All authors read and approved the final manuscript Publisher s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations Received: 1 November 17 Accepted: 7 May 18 References 1 Beesack, P: Hardy s inequality and its extension Pac J Math 11, ) Hall, R: Generalized Wirtinger inequalities, random matrix theory, and the zeros of the Riemann zeta-function J Number Theory 97, ) Lavoie, J, Osler, T, Tremblay, R: Fractional derivatives and special functions SIAM Rev 18), ) 4 Watson, G: A Treatise on the Theory of Bessel Functions Cambridge University Press, Cambridge 19)