An extension of Wirtinger's inequality to the complex integral

Sever Dragomir


In this paper we establish a natural extension of the Wirtinger inequality to the case of complex integral of analytic functions. Applications related to the trapezoid inequalities are also provided. Examples for logarithmic and exponential complex functions are given as well.


Wirtinger's inequality, Trapezoid inequality, Complex integral, Analytic functions

Full Text:



Alomari, M. W., textit{On Beesack--Wirtinger inequality},

Results Math., Online First 2017 Springer International Publishing DOI


Beesack, P.R., textit{Extensions of Wirtinger's inequality},

Trans. R. Soc. Can., textbf{53}(1959), 21-30.

Cerone, P., Dragomir, S.S., textit{A refinement of the Gr"{u}ss inequality and applications},

Tamkang J. Math., textbf{38}(2007), no. 1,

-49. Preprint RGMIA Res. Rep. Coll., textbf{5}(2)(2002),

Article 14. [Online: texttt{}].

Diaz, J.B., Metcalf, F.T., textit{Variations on Wirtinger's inequalit}y,

in: "Inequalities" Academic Press, New York, 1967, 79-103.

Fej'{e}r, L., textit{"{U}ber die Fourierreihen}, II, (Hungarian),

Math. Naturwiss, Anz. Ungar. Akad. Wiss., textbf{24}(1906),


Giova, R., {it An estimate for the best constant in the $L_{p}$-Wirtinger inequality with weights}, J. Func. Spaces Appl., textbf{6}(2008), no. 1, 1-16.

Jarov{s}, J., textit{On an integral inequality of the

Wirtinger type}, Appl. Math. Letters, textbf{24}(2011), 1389-1392.

Lee, C.F., Yeh, C.C., Hong, C.H., Agarwal, R.P., textit{Lyapunov and Wirtinger inequalities}, Appl. Math. Lett., textbf{17}(2004),


Ricciardi, T., textit{A sharp weighted Wirtinger inequality},

Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat., textbf{8}(1)(2005),


Swanson, C.A., textit{Wirtinger's inequality},

SIAM J. Math. Anal., textbf{9}(1978), 484-491.



  • There are currently no refbacks.