- Open Access
An investigation on a new class of harmonic mappings
© Yavuz Duman et al.; licensee Springer. 2013
- Received: 28 December 2012
- Accepted: 28 August 2013
- Published: 7 November 2013
In the present paper, we give an extension of the idea which was introduced by Sakaguchi (J. Math. Soc. Jpn. 11:72-75, 1959), and we give some applications of this extended idea for the investigation of the class of harmonic mappings.
- harmonic mappings
- distortion theorem
- grow theorem
Equivalently, the inequality holds for all . The family of all functions with the additional property that (i.e., ) is denoted by . Observe that the classical family of univalent functions consists of all functions such that for all . Thus, it is clear that .
Let Ω be the family of functions regular in the open unit disc and satisfy the conditions , for all .
for some and every .
Let and be analytic functions in . If there exists a function such that for every , then we say that is subordinate to and we write . We also note that if , then [2, 3]. We also note that is the analytic second dilatation of f and for every .
For this aim we need the following theorems and lemmas.
Theorem 1.1 
for all and .
Lemma 1.2 
Let be regular in the unit disc with , then if attains its maximum value on the circle at the point , one has for some .
Lemma 1.3 
If and are regular in , , maps onto a many-sheeted region which is starlike with respect to the origin, and , then .
Lemma 1.4 
for all .
for some and all . Since , we have that (2.1) is true. The sufficient part of the proof can be seen by following the above steps in the opposite direction by considering the subordination principle. □
Theorem 2.2 Let be an element of . If , then for all .
since and . But this is a contradiction to the condition , and so the assumption is wrong, i.e., for all . □
Remark 2.3 Theorem 2.2 is an extension of Lemma 1.3 to the harmonic mappings.
where is given by (1.6) for all and .
where is given by (1.6) for all , and is the Jacobian of f defined by for all .
Proof The proof of this corollary is a simple consequence of Lemma 1.1 and Lemma 1.4. □
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