Results & Lemmas (7)
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THEOREM 1.
THEOREM 1. For each 8 e U, /(/) | / E s F, 0) = f " f F * G(x) dx = eie). IZ Jo -1 Moroever ^C/)—75 e This result appears to be useful…
THEOREM 1. For each 8 e U, {/(/) | / E s{F, 0)} = f " f F * G(x) dx\zeAorz= eie). IZ Jo -1 Moroever ^C/)—75 e This result appears to be useful because it gives a complete description of the range of a linear operator over a class of functions defined by subordination and a constraint, namely the value of the angular limit at some point of the unit circle. Our conclusion will include some applications of our theorem. REMARK 1. It is clear that any bounded function in //(A) is integrable over [0,1
LEMMA 1.
LEMMA 1. Let g, h e H(A) be convex univalent. Then g*h is also convex univalent and fes(g)^f*hes(g*h). We say that feH(A) is properly…
LEMMA 1. Let g, h e H(A) be convex univalent. Then g*h is also convex univalent and fes(g)^f*hes(g*h). We say that feH(A) is properly subordinate to geH(A.) if f(z) = g(p(z)) where p e H(A) and \p(z)\ < \z\ if z =£0; we shall denote this by f<pg. In fact Ruscheweyh and Sheil-Small proved in [7] f*h<pg*h if f<pg and h, g are convex univalent. (1) We shall also need the following classical result, due to Julia (see [1, Section 1.4])
LEMMA 2.
LEMMA 2. Let v e H(A) with (z) ^ for all z e A. Then lim — exists and is 2-.1 ~Z either ^ 1 or infinite. Moreover if lim —: = / < °°, then…
LEMMA 2. Let v e H(A) with \v(z)\ ^ \z\ for all z e A. Then lim — exists and is 2-.1 \~Z either ^ 1 or infinite. Moreover if lim — : = / < °°, then 1 < lim v'(z) = I and equality is possible only if v(z) = z. z—\-z 2—i The next lemma appears in [4]. https://doi.org/10.1017/S0017089500009472 Published online by Cambridge University Press
LEMMA 3.
LEMMA 3. Let a.fieU. There exists a sequence of functions pk analytic in the closed unit disc such that (z) ^, pk(l) = e'a and lim pk z) =…
LEMMA 3. Let a.fieU. There exists a sequence of functions {pk} analytic in the closed unit disc such that \pk(z)\^\z\, pk(l) = e'a and lim pk{z) = e'^z uniformly on compact subsets of A. Ar~*°° If £ is a subset of C let E designate the closure of E. We shall finally need the following result, a proof of which, due to St. Ruscheweyh, is given below.
LEMMA 4.
LEMMA 4. Let f e H(A) be univalent, convex and bounded; let ThengjA)czf(A). The fact that U(z) = —z log(l — z) is convex univalent in A…
LEMMA 4. Let f e H(A) be univalent, convex and bounded; let ThengjA)czf(A). The fact that U(z) = —z log(l — z) is convex univalent in A will be used at several stages of our work.
COROLLARY 1. · coeff
COROLLARY 1. Let A e [0, 1] and px denote the order of starlikeness of Sk. Then 3° Finally we remark that Theorem 1 can be used to obtain a…
COROLLARY 1. Let A e [0, 1] and px denote the order of starlikeness of Sk. Then 3° Finally we remark that Theorem 1 can be used to obtain a sharp variability region for certain combinations of the Taylor coefficients of functions / e s(F, 9). For oo example if F(z) = S Anz" e //(A) is convex univalent and bounded, we obtain the n = l following result as an immediate consequence of Theorem 1, by choosing G(z) = z + n-2zn.
COROLLARY 2.
COROLLARY 2. For any 6 eU, we have REFERENCES 1. L. Ahlfors, Conformal invariants (McGraw-Hill, 1973). 2. L. Brickman, T. H. MacGregor and…
COROLLARY 2. For any 6 eU, we have REFERENCES 1. L. Ahlfors, Conformal invariants (McGraw-Hill, 1973). 2. L. Brickman, T. H. MacGregor and D. R. Wilken, Convex hulls of some classical families of univalent functions, Trans. Amer. Math. Soc, 156 (1971), 91-107. 3. P. L. Duren, Univalent functions (Springer Verlag, 1983). 4. R. Fournier, On integrals of bounded analytic functions in the closed unit disc, Complex Variables, 11 (1989), 125-133. 5. D. J. Hallenbeck and T. H. MacGregor, Linear problem
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