In this paper, vector norm inequalities that provides upper bounds for the Lipschitz quantity
║f (T) x - f (V ) x ║ for power series f(z) = ∑∞n=0 anzn; bounded linear operators T; V on the
Hilbert space H and vectors x ∈ H are established. Applications in relation to Hermite-
Hadamard type inequalities and examples for elementary functions of interest are given as
 H. Araki and S. Yamagami, An inequality for Hilbert-Schmidt norm, Commun. Math. Phys.
81 (1981), 89-96.
 R. Bhatia, First and second order perturbation bounds for the operator absolute value, Linear
Algebra Appl. 208/209 (1994), 367-376.
 R. Bhatia, Perturbation bounds for the operator absolute value. Linear Algebra Appl. 226/228
 R. Bhatia, D. Singh and K. B. Sinha, Differentiation of operator functions and perturbation
bounds. Comm. Math. Phys. 191 (1998), no. 3, 603-611.
 R. Bhatia, Matrix Analysis, Springer Verlag, 1997.
 L. Ciurdariu, A note concerning several Hermite-Hadamard inequalities for different types of
convex functions. Int. J. Math. Anal. 6 (2012), no. 33-36, 1623-1639.
 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 (2000), no. 2, 489-501.
 S. S. Dragomir and C. E. M. Pearce, Selected Topics on Hermite-
Hadamard Inequalities and Applications, RGMIA Monographs, 2000. [Online
 S. S. Dragomir, Inequalities of Lipschitz type for power series of operators in Hilbert spaces,
Commun. Math. Anal. 16 (2014), Number 1, 102 - 122.
 R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972.
 Yu. B. Farforovskaya, Estimates of the closeness of spectral decompositions of self-adjoint op-
erators in the Kantorovich-Rubinshtein metric (in Russian), Vesln. Leningrad. Gos. Univ. Ser.
Mat. Mekh. Astronom. 4 (1967), 155-156.
 Yu. B. Farforovskaya, An estimate of the norm ||f (B) - f (A)|| for self-adjoint operators A
and B (in Russian) Zap. Nauchn. Sem. Leningrad. Otdel. Mat. Inst. 56 (1976), 143-162 .
 Yu. B. Farforovskaya and L. Nikolskaya, Modulus of continuity of operator functions. Algebra
i Analiz 20 (2008), no. 3, 224-242; translation in St. Petersburg Math. J. 20 (2009), no. 3,
 Y. Feng and W. Zhao, Renement of Hermite-Hadamard inequality. Far East J. Math. Sci. 68
(2012), no. 2, 245-250.
 X. Gao, A note on the Hermite-Hadamard inequality. J. Math. Inequal. 4 (2010), no. 4, 587-
 S.-R. Hwang, K.-L. Tseng and K.-C. Hsu, Hermite-Hadamard type and Fejer type inequalities
for general weights (I). J. Inequal. Appl. 2013, 2013:170.
 T. Kato, Continuity of the map S ! jSj for linear operators, Proc. Japan Acad. 49 (1973),
 U. S. Krmac and R. Dikici, On some Hermite-Hadamard type inequalities for twice differen-
tiable mappings and applications. Tamkang J. Math. 44 (2013), no. 1, 41-51.
 J. Mikusinski, The Bochner Integral, Birkhauser Verlag, 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 (2012), no. 1,
 M. Matic and J. Pecaric, Note on inequalities of Hadamard's type for Lipschitzian mappings.
Tamkang J. Math. 32 (2001), no. 2, 127-130.
 W. Rudin, Functional Analysis, McGraw Hill, 1973.
 M. Z. Sarikaya, On new Hermite Hadamard Fejer type integral inequalities. Stud. Univ. Babes-
Bolyai Math. 57 (2012), no. 3, 377-386.
 S. Wasowicz and A. Witkowski, On some inequality of Hermite-Hadamard type. Opuscula
Math. 32 (2012), no. 3, 591-600.
 B.-Y. Xi and F. Qi, Some integral inequalities of Hermite-Hadamard type for convex functions
with applications to means. J. Funct. Spaces Appl. 2012, Art. ID 980438, 14 pp.
 G. Zabandan, A. Bodaghi and A. Klcman, The Hermite-Hadamard inequality for r-convex
functions. J. Inequal. Appl. 2012, 2012:215, 8 pp.
 C.-J. Zhao, W.-S. Cheung and X.-Y. Li, On the Hermite-Hadamard type inequalities. J. Inequal.
Appl. 2013, 2013:228.