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Integral Means of Functions with Positive Real Part

Published online by Cambridge University Press:  20 November 2018

F. Holland  and
J. B. Twomev
F. Holland
Affiliation:
University College, Cork, Ireland
J. B. Twomev
Affiliation:
University College, Cork, Ireland
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We denote by the class of functions of the form

that are regular in Δ = {z:|;z| < 1} and satisfy Re h(z) > 0 there. For 0 ≦ r < 1, we write

We note that, for , the inequality

is classical.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 32 , Issue 4 , April 1980 , pp. 1008 - 1020
Copyright
Copyright © Canadian Mathematical Society 1980

References

1. Duren, P. L., Theory of Hp spaces (Academic Press, 1970).Google Scholar
2. Hardy, G. H. and Littlewood, J. E., Theorems concerning mean values of analytic or harmonic functions, Quart. J. Math. 12 (1941), 221–256.Google Scholar
3. Hardy, G. H. and Littlewood, J. E., Some properties of fractional integrals II, Math. Z. 34 (1931), 403–439.Google Scholar
4. Hayman, W. K., On functions with positive real part, J. London Math. Soc. 36 (1961), 35–48.Google Scholar
5. Keogh, F. R., Some theorems on conformai mapping of bounded star-shaped domains, Proc. London Math. Soc. 9 (1959), 481–491.Google Scholar
6. Lewis, J. L., Note on an arc length problem, J. London Math. Soc. 12 (1976), 469–474.Google Scholar
7. Macintyre, A. J. and Rogosinski, W. W., Some elementary inequalities in function theory, Edinburgh Math. Notes 35 (1945), 1–3.Google Scholar
8. Noonan, J. W. and Thomas, D. K., The integral means of regular functions, J. London Math. Soc. 9 (1975), 557–560.Google Scholar
9. Rudin, W., Real and complex analysis, Second Edition (McGraw-Hill, 1974).Google Scholar
10. Salem, R., On a theorem of Zygmund, Duke Math. J. 10 (1943), 23–31.Google Scholar
11. Twomey, J. B., On bounded starlike functions, J. Analyse Mathématique 24 (1971). 191–204.Google Scholar
12. Zygmund, A., Trigonometric series, Vol. I. (University Press, 1959).Google Scholar