Research Article

### Ostrowski's Type Inequalities for the Complex Integral on Paths

Year 2020, Volume 3, Issue 4, 125 - 138, 01.12.2020

### Abstract

In this paper we extend the Ostrowski inequality to the integral with respect to arc-length by providing upper bounds for the quantity |f(v)ℓ(γ)-∫_{γ}f(z)|dz|| under the assumptions that γ is a smooth path parametrized by z(t), t∈[a,b] with the length ℓ(γ), u=z(a), v=z(x) with x∈(a,b) and w=z(b) while f is holomorphic in G, an open domain and γ⊂G. An application for circular paths is also given. Several applications for circular paths and for some special functions of interest such as the exponential functions are also provided.

### References

• S. S. Dragomir: A refinement of Ostrowski's inequality for absolutely continuous functions whose derivatives belong to $L_{\infty }$ and applications. Libertas Math. 22 (2002), 49--63.
• S. S. Dragomir: A refinement of Ostrowski's inequality for absolutely continuous functions and applications. Acta Math. Vietnam 27 (2002), no. 2, 203--217.
• S. S. Dragomir: A functional generalization of Ostrowski inequality via Montgomery identity. Acta Math. Univ. Comenian. (N.S.) 84 (2015), no. 1, 63--78. Preprint RGMIA Res. Rep. Coll. 16 (2013), Art. 65, pp. 15 [Online http://rgmia.org/papers/v16/v16a65.pdf].
• S. S. Dragomir: Ostrowski type inequalities for Lebesgue integral: a survey of recent results. Aust. J. Math. Anal. Appl. 14 (2017), no. 1, Art. 1, 283 pp.
• S. S. Dragomir: An extension of Ostrowski's inequality to the complex integral}. Preprint RGMIA Res. Rep. Coll. 18 (2018), Art. 112, 17 pp. [Online https://rgmia.org/papers/v21/v21a112.pdf].
• S. S. Dragomir, S. Wang: Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules. Appl. Math. Lett. 11 (1) (1998), 105-109.
• D. S. Mitrinovi\'{c}, J. E. Pe\v{c}ari\'{c} and A. M. Fink: Inequalities for Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht, 1994.
• A. Ostrowski: \"{U}ber die Absolutabweichung einerdifferentiierbaren Funktion von ihrem Integralmittelwert}. Comment. Math. Helv. 10 (1938), 226-227.
Year 2020, Volume 3, Issue 4, 125 - 138, 01.12.2020

### References

• S. S. Dragomir: A refinement of Ostrowski's inequality for absolutely continuous functions whose derivatives belong to $L_{\infty }$ and applications. Libertas Math. 22 (2002), 49--63.
• S. S. Dragomir: A refinement of Ostrowski's inequality for absolutely continuous functions and applications. Acta Math. Vietnam 27 (2002), no. 2, 203--217.
• S. S. Dragomir: A functional generalization of Ostrowski inequality via Montgomery identity. Acta Math. Univ. Comenian. (N.S.) 84 (2015), no. 1, 63--78. Preprint RGMIA Res. Rep. Coll. 16 (2013), Art. 65, pp. 15 [Online http://rgmia.org/papers/v16/v16a65.pdf].
• S. S. Dragomir: Ostrowski type inequalities for Lebesgue integral: a survey of recent results. Aust. J. Math. Anal. Appl. 14 (2017), no. 1, Art. 1, 283 pp.
• S. S. Dragomir: An extension of Ostrowski's inequality to the complex integral}. Preprint RGMIA Res. Rep. Coll. 18 (2018), Art. 112, 17 pp. [Online https://rgmia.org/papers/v21/v21a112.pdf].
• S. S. Dragomir, S. Wang: Applications of Ostrowski's inequality to the estimation of error bounds for some special means and for some numerical quadrature rules. Appl. Math. Lett. 11 (1) (1998), 105-109.
• D. S. Mitrinovi\'{c}, J. E. Pe\v{c}ari\'{c} and A. M. Fink: Inequalities for Functions and Their Integrals and Derivatives. Kluwer Academic Publishers, Dordrecht, 1994.
• A. Ostrowski: \"{U}ber die Absolutabweichung einerdifferentiierbaren Funktion von ihrem Integralmittelwert}. Comment. Math. Helv. 10 (1938), 226-227.

### Details

Primary Language English Mathematics Articles Sever DRAGOMIR> (Primary Author) Victoria University 0000-0003-2902-6805 Australia December 1, 2020 Year 2020, Volume 3, Issue 4

### Cite

 Bibtex @research article { cma798861, journal = {Constructive Mathematical Analysis}, issn = {2651-2939}, address = {}, publisher = {Tuncer ACAR}, year = {2020}, volume = {3}, number = {4}, pages = {125 - 138}, doi = {10.33205/cma.798861}, title = {Ostrowski's Type Inequalities for the Complex Integral on Paths}, key = {cite}, author = {Dragomır, Sever} } APA Dragomır, S. (2020). Ostrowski's Type Inequalities for the Complex Integral on Paths . Constructive Mathematical Analysis , 3 (4) , 125-138 . DOI: 10.33205/cma.798861 MLA Dragomır, S. "Ostrowski's Type Inequalities for the Complex Integral on Paths" . Constructive Mathematical Analysis 3 (2020 ): 125-138 Chicago Dragomır, S. "Ostrowski's Type Inequalities for the Complex Integral on Paths". Constructive Mathematical Analysis 3 (2020 ): 125-138 RIS TY - JOUR T1 - Ostrowski's Type Inequalities for the Complex Integral on Paths AU - SeverDragomır Y1 - 2020 PY - 2020 N1 - doi: 10.33205/cma.798861 DO - 10.33205/cma.798861 T2 - Constructive Mathematical Analysis JF - Journal JO - JOR SP - 125 EP - 138 VL - 3 IS - 4 SN - 2651-2939- M3 - doi: 10.33205/cma.798861 UR - https://doi.org/10.33205/cma.798861 Y2 - 2020 ER - EndNote %0 Constructive Mathematical Analysis Ostrowski's Type Inequalities for the Complex Integral on Paths %A Sever Dragomır %T Ostrowski's Type Inequalities for the Complex Integral on Paths %D 2020 %J Constructive Mathematical Analysis %P 2651-2939- %V 3 %N 4 %R doi: 10.33205/cma.798861 %U 10.33205/cma.798861 ISNAD Dragomır, Sever . "Ostrowski's Type Inequalities for the Complex Integral on Paths". Constructive Mathematical Analysis 3 / 4 (December 2020): 125-138 . https://doi.org/10.33205/cma.798861 AMA Dragomır S. Ostrowski's Type Inequalities for the Complex Integral on Paths. CMA. 2020; 3(4): 125-138. Vancouver Dragomır S. Ostrowski's Type Inequalities for the Complex Integral on Paths. Constructive Mathematical Analysis. 2020; 3(4): 125-138. IEEE S. Dragomır , "Ostrowski's Type Inequalities for the Complex Integral on Paths", Constructive Mathematical Analysis, vol. 3, no. 4, pp. 125-138, Dec. 2020, doi:10.33205/cma.798861