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Harmonic function for which the second dilatation is α-spiral
Journal of Inequalities and Applications volume 2012, Article number: 262 (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 .
Lewy  proved in 1936 that the harmonic function f is locally univalent in a simply connected domain if and only if its Jacobian
is different from zero in . In view of this result, locally univalent harmonic mappings in the unit disc are either sense-reversing if
in or sense-preserving if
in . Throughout this paper, we will restrict ourselves to the study of sense-preserving harmonic mappings. However, since f is sense-preserving if and only if is sense-reserving, all the results obtained in this article regarding sense-preserving harmonic mappings can be adapted to sense-reversing ones. Note that is sense-preserving in if and only if does not vanish in the unit disc and the second-complex dilatation has the property in ; therefore, we can take , . Thus, the class of all harmonic mappings being sense-preserving in the unit disc can be defined by
Let Ω be the family of functions which are regular in and satisfy the conditions , for all . Denote by P the family of functions which are regular in such that
for some function for all .
Next, let denote the family of functions which are regular in such that
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 .
Now, we consider the following class of harmonic mappings in the plane:
In the present paper, we will investigate the class .
We will need the following lemma and theorem in the sequel.
Let be an element of , then
for all .
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
Lemma 2.1 Let be an element of then
for all . This inequality is sharp because the extremal function is
then the function
satisfies the condition of the Schwarz lemma. Using the definition of subordination, we have
On the other hand, the transformation maps onto the disc with the center
and the radius
Therefore, we can write
which gives (2.1). □
Corollary 2.2 Let , then
for all .
Using Lemma 1.2 and Lemma 2.1, then we can write
If we use Theorem 1.1 in the inequalities (2.5) and (2.6), we get (2.3) and (2.4). □
Corollary 2.3 Let be an element of , then
for all .
using Lemma 2.1 and Theorem 1.1 in the equality (2.8) and after simple calculations, we get (2.7). □
Corollary 2.4 If is an element of , then
where for all .
Using Corollary 2.2 and Theorem 1.1, we obtain
Therefore, we have
Integrating the last inequality (2.10), we get (2.9). □
Theorem 2.5 Let be an element of , then
where ; ; and are the coefficients of the functions and ; ; ; .
We denote by
then we have
Since is in P, there is a function satisfying the conditions of the Schwarz lemma such that
Using this equation in (2.12) and after the following calculations given above
we get the following equality:
If , we have
Therefore, the equality (2.15) can be considered in the following form:
Using the Clunie method , then we can write
Eventually, we will let , then we have
Duren P: Harmonic Mapping in the Plane. Cambridge Press, Cambridge; 2004.
Lewy H: On the non-vanishing of the Jacobian in certain in one-to-one mappings. Bull. Am. Math. Soc. 1936, 42: 689–692. 10.1090/S0002-9904-1936-06397-4
Goodman AW I. In Univalent Functions. Mariner Publishing Company, Tampa; 1983.
Goodman AW II. In Univalent Functions. Mariner Publishing Company, Tampa; 1983.
Bernardi SD: Convex and starlike univalent functions. Trans. Am. Math. Soc. 1969, 135: 429–446.
Clunie J: On meromorphic Schlicht functions. J. Lond. Math. Soc. 1959, 34: 215–216. 10.1112/jlms/s1-34.2.215
The authors declare that they have no competing interests.
All authors contributed equally and significantly to writing this paper. All authors read and approved the final manuscript.
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Aydog̃an, M., Duman, E.Y., Polatog̃lu, Y. et al. Harmonic function for which the second dilatation is α-spiral. J Inequal Appl 2012, 262 (2012). https://doi.org/10.1186/1029-242X-2012-262
- Harmonic functions
- growth theorem
- distortion theorem
- coefficient inequality