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} a_{n}z^{n}; 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
well.

[1] H. Araki and S. Yamagami, An inequality for Hilbert-Schmidt norm, Commun. Math. Phys. 81 (1981), 89-96.

[2] R. Bhatia, First and second order perturbation bounds for the operator absolute value, Linear Algebra Appl. 208/209 (1994), 367-376.

[3] R. Bhatia, Perturbation bounds for the operator absolute value. Linear Algebra Appl. 226/228 (1995), 639-645.

[4] R. Bhatia, D. Singh and K. B. Sinha, Differentiation of operator functions and perturbation bounds. Comm. Math. Phys. 191 (1998), no. 3, 603-611.

[5] R. Bhatia, Matrix Analysis, Springer Verlag, 1997.

[6] 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.

[7] 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.

[8] 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].

[9] S. S. Dragomir, Inequalities of Lipschitz type for power series of operators in Hilbert spaces, Commun. Math. Anal. 16 (2014), Number 1, 102 - 122.

[10] R. Douglas, Banach Algebra Techniques in Operator Theory, Academic Press, 1972.

[11] 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.

[12] 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 .

[13] 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, 493-506.

[14] Y. Feng and W. Zhao, Renement of Hermite-Hadamard inequality. Far East J. Math. Sci. 68 (2012), no. 2, 245-250.

[15] X. Gao, A note on the Hermite-Hadamard inequality. J. Math. Inequal. 4 (2010), no. 4, 587- 591.

[16] 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.

[17] T. Kato, Continuity of the map S ! jSj for linear operators, Proc. Japan Acad. 49 (1973), 143-162.

[18] 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.

[19] J. Mikusinski, The Bochner Integral, Birkhauser Verlag, 1978.

[20] 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, 9-17.

[21] M. Matic and J. Pecaric, Note on inequalities of Hadamard's type for Lipschitzian mappings. Tamkang J. Math. 32 (2001), no. 2, 127-130.

[22] W. Rudin, Functional Analysis, McGraw Hill, 1973.

[23] M. Z. Sarikaya, On new Hermite Hadamard Fejer type integral inequalities. Stud. Univ. Babes- Bolyai Math. 57 (2012), no. 3, 377-386.

[24] S. Wasowicz and A. Witkowski, On some inequality of Hermite-Hadamard type. Opuscula Math. 32 (2012), no. 3, 591-600.

[25] 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.

[26] G. Zabandan, A. Bodaghi and A. Klcman, The Hermite-Hadamard inequality for r-convex functions. J. Inequal. Appl. 2012, 2012:215, 8 pp.

[27] C.-J. Zhao, W.-S. Cheung and X.-Y. Li, On the Hermite-Hadamard type inequalities. J. Inequal. Appl. 2013, 2013:228.