We obtain some inequalities of the Hermite–Hadamard type for *K*-bounded norm-convex mappings between two normed spaces. The applications for twice differentiable functions in Banach spaces and functions defined by power series in Banach algebras are presented. Some discrete Jensen-type inequalities are also obtained.

This is a preview of subscription content, access via your institution.

## References

G. A. Anastassiou, “Univariate Ostrowski inequalities, revisited,”

*Monatsh. Math.*,**135**, No. 3, 175–189 (2002).H. Araki and S. Yamagami, “An inequality for Hilbert–Schmidt norm,”

*Comm. Math. Phys.*,**81**, 89–96 (1981).R. Bhatia, “First and second order perturbation bounds for the operator absolute value,”

*Linear Algebra Appl.*,**208/209**, 367–376 (1994).R. Bhatia, “Perturbation bounds for the operator absolute value,”

*Linear Algebra Appl.*,**226/228**, 639–645 (1995).R. Bhatia, D. Singh, and K. B. Sinha, “Differentiation of operator functions and perturbation bounds,”

*Comm. Math. Phys.*,**191**, No. 3, 603–611 (1998).R. Bhatia,

*Matrix Analysis*, Springer Verlag (1997).P. Cerone and S. S. Dragomir, “Midpoint-type rules from an inequalities point of view,”

*Handb. Anal.-Comput. Meth. Appl. Math.*, Ed. G. A. Anastassiou, CRC Press, New York, 135–200 (2003).P. Cerone and S. S. Dragomir, “New bounds for the three-point rule involving the Riemann–Stieltjes integrals,”

*Adv. Stat. Combin. Related Areas*, World Scientific, (2002), pp. 53–62.P. Cerone, S. S. Dragomir, and J. Roumeliotis, “Some Ostrowski-type inequalities for

*n*-time differentiable mappings and applications,”*Demonstr. Math.*,**32**, No. 2, 697–712 (1999).L. Ciurdariu, “A note concerning several Hermite–Hadamard inequalities for different types of convex functions,”

*Int. J. Math. Anal.*,**6**, No. 33–36, 1623–1639 (2012).R. Coleman,

*Calculus on Normed Vector Spaces*, Springer, New York, etc. (2012).S. S. Dragomir, Y. J. Cho, and S. S. Kim, “Inequalities of Hadamard’s type for Lipschitzian mappings and their applications,”

*J. Math. Anal. Appl.*,**245**, No. 2, 489–501 (2000).S. S. Dragomir and C. E. M. Pearce, “Selected topics on Hermite–Hadamard inequalities and applications,”

*RGMIA Monographs*(2000) [Online http://rgmia.org/monographs/hermite_hadamard.html].S. S. Dragomir,

*Semi-Inner Products and Applications*, Nova Science Publ., Hauppauge, NY (2004), x+222 p.S. S. Dragomir, “Ostrowski’s inequality for monotonous mappings and applications,”

*J. KSIAM*,**3**, No. 1, 127–135 (1999).S. S. Dragomir, “The Ostrowski’s integral inequality for Lipschitzian mappings and applications,”

*Comput. Math. Appl.*,**38**, 33–37 (1999).S. S. Dragomir, “On the Ostrowski’s inequality for Riemann–Stieltjes integral,”

*Korean J. Appl. Math.*,**7**, 477–485 (2000).S. S. Dragomir, “On the Ostrowski’s inequality for mappings of bounded variation and applications,”

*Math. Inequal. Appl.*,**4**, No. 1, 33–40 (2001).S. S. Dragomir, “On the Ostrowski inequality for Riemann–Stieltjes integral \( {\int}_a^bf(t) du(t) \)

*,*where*f*is of Hölder type and*u*is of bounded variation and applications,”*J. KSIAM*,**5**, No. 1, 35–45 (2001).S. S. Dragomir, “Ostrowski-type inequalities for isotonic linear functionals,”

*J. Inequal. Pure Appl. Math.*,**3**, No. 5, Art. 68 (2002).S. S. Dragomir, “An inequality improving the first Hermite–Hadamard inequality for convex functions defined on linear spaces and applications for semi-inner products,”

*J. Inequal. Pure Appl. Math.*,**3**, No. 2, Art. 31 (2002).S. S. Dragomir, “An Ostrowski like inequality for convex functions and applications,”

*Rev. Mat. Comput.*,**16**, No. 2, 373–382 (2003).S. S. Dragomir, “Operator inequalities of Ostrowski and trapezoidal type,”

*Springer Briefs Math.*, Springer, New York (2012).S. S. Dragomir, “Inequalities for power series in Banach algebras,”

*SUT J. Math.*,**50**, No. 1, 25–45 (2014).S. S. Dragomir, “Integral inequalities for Lipschitzian mappings between two Banach spaces and applications,”

*Kodai Math. J.*,**39**, 227–251 (2016).S. S. Dragomir, P. Cerone, J. Roumeliotis, and S. Wang, “A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis,”

*Bull. Math. Soc. Sci. Math. Roumanie (N.S.)*,**42(90)**, No. 4, 301–314 (1999).S. S. Dragomir and Th. M. Rassias (Eds),

*Ostrowski Type Inequalities and Applications in Numerical Integration*, Kluwer Academic Publ. (2002).S. S. Dragomir and S. Wang, “A new inequality of Ostrowski’s type in

*L*_{1}-norm and applications to some special means and to some numerical quadrature rules,”*Tamkang J. Math.*,**28**, 239–244 (1997).S. S. Dragomir and S. Wang, “Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and some numerical quadrature rules,”

*Appl. Math. Lett.*,**11**, 105–109 (1998).S. S. Dragomir and S. Wang, “A new inequality of Ostrowski’s type in

*L*_{ p }-norm and applications to some special means and to some numerical quadrature rules,”*Indian J. Math.*,**40**, No. 3, 245–304 (1998).R. Douglas,

*Banach Algebra Techniques in Operator Theory*, Academic Press (1972).Yu. B. Farforovskaya, “Estimates for the closeness of spectral decompositions of self-adjoint operators in the Kantorovich–Rubinshtein metric,”

*Vestn. Leningrad. Gos. Univ., Ser. Mat., Mekh., Astronom.*,**4**, 155–156 (1967).Yu. B. Farforovskaya, “An estimate of the norm ‖

*f*(*B*) −*f*(*A*)‖ for self-adjoint operators*A*and*B*,”*Zap. Nauch. Sem. Leningrad. Otdel. Mat. Inst.*,**56**, 143–162 (1976).Yu. B. Farforovskaya and L. Nikol’skaya, “Modulus of continuity of operator functions,”

*Algebra Anal.*,**20**, No. 3, 224–242 (2008);*English translation: St. Petersburg Math. J.*,**20**, No. 3, 493–506 (2009).Y. Feng and W. Zhao, “Refinement of Hermite–Hadamard inequality,”

*Far East J. Math. Sci.*,**68**, No. 2, 245–250 (2012).A. M. Fink, “Bounds on the deviation of a function from its averages,”

*Czechoslovak Math. J.*,**42(117)**, No. 2, 298–310 (1992).X. Gao, “A note on the Hermite–Hadamard inequality,”

*J. Math. Inequal.*,**4**, No. 4, 587–591 (2010).S.-R. Hwang, K.-L. Tseng, and K.-C. Hsu, “Hermite–Hadamard type and Fejér type inequalities for general weights (I),”

*J. Inequal. Appl.*,**2013**, No. 170 (2013).T. Kato, “Continuity of the map

*S*→ |*S*| for linear operators,”*Proc. Jap. Acad.*,**49**, 143–162 (1973).U. S. Kırmacı and R. Dikici, “On some Hermite–Hadamard type inequalities for twice differentiable mappings and applications,”

*Tamkang J. Math.*,**44**, No. 1, 41–51 (2013).J. Mikusínski,

*The Bochner Integral*, Birkhäuser, Basel (1978).M. Muddassar, M. I. Bhatti, and M. Iqbal, “Some new

*s*-Hermite–Hadamard type inequalities for differentiable functions and their applications,”*Proc. Pakistan Acad. Sci.*,**49**, No. 1, 9–17 (2012).M. Matić and J. Pečarić, “Note on inequalities of Hadamard’s type for Lipschitzian mappings,”

*Tamkang J. Math.*,**32**, No. 2, 127–130 (2001).A. Ostrowski, “Über die Absolutabweichung einer differentienbaren Funktionen von ihren Integralmittelwert,”

*Comment. Math. Helv.*,**10**, 226–227 (1938).W. Rudin,

*Functional Analysis*, McGraw Hill (1973).M. Z. Sarikaya, “On new Hermite–Hadamard-Fejér type integral inequalities,”

*Stud. Univ. Babeş-Bolyai Math.*,**57**, No. 3, 377–386 (2012).S. Wąsowicz and A. Witkowski, “On some inequality of Hermite–Hadamard type,”

*Opusc. Math.*,**32**, No. 3, 591–600 (2012).B.-Y. Xi and F. Qi, “Some integral inequalities of Hermite–Hadamard type for convex functions with applications to means,”

*J. Funct. Spaces Appl.*, Art. ID 980438, 14 p. (2012).G. Zabandan, A. Bodaghi, and A. Kılıçman, “The Hermite–Hadamard inequality for

*r*-convex functions,”*J. Inequal. Appl.*,**2012**, No. 215, 8 p. (2012).C.-J. Zhao, W.-S. Cheung, and X.-Y. Li, “On the Hermite–Hadamard type inequalities,”

*J. Inequal. Appl.*,**2013**, No. 228 (2013).

## Author information

### Authors and Affiliations

## Additional information

Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 68, No. 10, pp. 1330–1347, October, 2016.

## Rights and permissions

## About this article

### Cite this article

Dragomir, S.S. Integral Inequalities of the Hermite–Hadamard Type for *K*-Bounded Norm-Convex Mappings.
*Ukr Math J* **68**, 1530–1551 (2017). https://doi.org/10.1007/s11253-017-1311-0

Received:

Published:

Issue Date:

DOI: https://doi.org/10.1007/s11253-017-1311-0