Harmonic function for which the second dilatation is α-spiral
© Aydoğan et al.; licensee Springer 2012
Received: 15 June 2012
Accepted: 24 September 2012
Published: 6 November 2012
Let be a harmonic function in the unit disc . We will give some properties of f under the condition the second dilatation is α-spiral.
A planar harmonic mapping in the unit disc is a complex-valued harmonic function f which maps onto some planar domain . Since is simply connected, the mapping f has a canonical decomposition , where h and g are analytic in . As usual, we call h the analytic part of f and g the co-analytic part of f. An elegant and complete account of the theory of planar harmonic mapping is given in Duren’s monograph .
for some function for all .
for some for all .
Let and be analytic functions in . If there exists such that for all , then we say that is subordinate to and we write , then .
In the present paper, we will investigate the class .
We will need the following lemma and theorem in the sequel.
These inequalities are sharp because the extremal function is .
Let and be regular in , map onto a many-sheeted starlike region, , for . . Then for .
2 Main results
which gives (2.1). □
for all .
If we use Theorem 1.1 in the inequalities (2.5) and (2.6), we get (2.3) and (2.4). □
for all .
using Lemma 2.1 and Theorem 1.1 in the equality (2.8) and after simple calculations, we get (2.7). □
where for all .
Integrating the last inequality (2.10), we get (2.9). □
where ; ; and are the coefficients of the functions and ; ; ; .
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