A New Look for Starlike Logharmonic Mappings
Author(s)
Zayid Abdulhadi
ABSTRACT
A function f(z) defined on the unit disc U is said to be logharmonic if it is the solution of the
nonlinear elliptic partial differential equation 
where 
such
that 
. These mappings admit a global
representation of the form
where 
In this paper,we shall
consider the logharmonic mappings 
, where 
is
starlike. Distortion theorem and radius of starlikess are obtained. Moreover,
we use star functions to determine the integral means for these mappings. An
upper bound for the arclength is included.
Cite this paper
Abdulhadi, Z. (2014) A New Look for Starlike Logharmonic Mappings. Applied Mathematics, 5, 1053-1060. doi: 10.4236/am.2014.57099.
Abdulhadi, Z. (2014) A New Look for Starlike Logharmonic Mappings. Applied Mathematics, 5, 1053-1060. doi: 10.4236/am.2014.57099.
References
[1] Abdulhadi, Z. (1996) Close-to-Starlike Logharmonic Mappings. The International Journal of Mathematics and Mathematical Sciences, 19, 563-574. [2] Abdulhadi, Z. (2002) Typically Real Logharmonic Mappings. The International Journal of Mathematics and Mathematical Sciences, 31, 1-9. [3] Abdulhadi, Z. and Bshouty, D. (1988) Univalent Functions in . Transactions of the AMS—American Mathematical Society, 305, 841-849. [4] Abdulhadi, Z. and Hengartner, W. (1987) Spirallike Logharmonic Mappings. Complex Variables, Theory and Application, 9, 121-130. [5] Abdulhadi, Z., Hengartner, W. and Szynal, J. (1993) Univalent Logharmonic Ring Mappings. Proceedings of the American Mathematical Society, 119, 735-745. [6] Abdulhadi, Z. and Hengartner, W. (1996) One Pointed Univalent Logharmonic Mappings. Journal of Mathematical Analysis and Applications, 203, 333-351. [7] Abdulhadi, Z. and Hengartner, W. (2001) Polynomials in . Complex Variables, Theory and Application, 46, 89107. [8] Abu-Muhanna, Y. and Lyzzaik, A. (1990) The Boundary Behaviour of Harmonic Univalent Maps. Pacific Journal of Mathematics, 141, 1-20.
http://dx.doi.org/10.2140/pjm.1990.141.1 [9] Clunie, J. and Sheil-Smal, T. (1984) Harmonic Univalent Functions. Annales Academic Scientiarum Fennice Mathematica, 9, 3-25. [10] Duren, P. and Schober, G. (1987) A Variational Method for Harmonic Mappings on Convex Regions. Complex Variables, Theory and Application, 9, 153-168.
http://dx.doi.org/10.1080/17476938708814259 [11] Duren, P. and Schober, G. (1989) Linear Extremal Problems for Harmonic Mappings of the Disk. Proceedings of the American Mathematical Society, 106, 967-973.
http://dx.doi.org/10.1090/S0002-9939-1989-0953740-5 [12] Hengartner, W. and Schober, G. (1986) On the Boundary Behavior of Orientation-Preserving Harmonic Mappings. Complex Variables, Theory and Application, 5, 197-208.
http://dx.doi.org/10.1080/17476938608814140 [13] Hengartner, W. and Schober, G. (1986) Harmonic Mappings with Given Dilatation. Journal London Mathematical Society, 33, 473-483.
http://dx.doi.org/10.1112/jlms/s2-33.3.473 [14] Jun, S.H. (1993) Univalent Harmonic Mappings on Proceedings of the American Mathematical Society, 119, 109-114.
http://dx.doi.org/10.1090/S0002-9939-1993-1148026-3 [15] Nitsche, J.C.C. (1989) Lectures on Minimal Surfaces. Vol. 1, Translated from the German by Jerry M. Feinberg, Cambridge University Press, Cambridge. [16] Osserman, R. (1986) A Survey of Minimal Surfaces. 2nd Edition, Dover, New York. [17] Baernstein, A. (1974) Integral Means, Univalent Functions and Circular Symmetrization. Acta Mathematica, 133, 139169.
http://dx.doi.org/10.1007/BF02392144 [18] Duren, P. (1983) Univalent Functions. Springer-Verlag, Berlin.
[1] Abdulhadi, Z. (1996) Close-to-Starlike Logharmonic Mappings. The International Journal of Mathematics and Mathematical Sciences, 19, 563-574. [2] Abdulhadi, Z. (2002) Typically Real Logharmonic Mappings. The International Journal of Mathematics and Mathematical Sciences, 31, 1-9. [3] Abdulhadi, Z. and Bshouty, D. (1988) Univalent Functions in . Transactions of the AMS—American Mathematical Society, 305, 841-849. [4] Abdulhadi, Z. and Hengartner, W. (1987) Spirallike Logharmonic Mappings. Complex Variables, Theory and Application, 9, 121-130. [5] Abdulhadi, Z., Hengartner, W. and Szynal, J. (1993) Univalent Logharmonic Ring Mappings. Proceedings of the American Mathematical Society, 119, 735-745. [6] Abdulhadi, Z. and Hengartner, W. (1996) One Pointed Univalent Logharmonic Mappings. Journal of Mathematical Analysis and Applications, 203, 333-351. [7] Abdulhadi, Z. and Hengartner, W. (2001) Polynomials in . Complex Variables, Theory and Application, 46, 89107. [8] Abu-Muhanna, Y. and Lyzzaik, A. (1990) The Boundary Behaviour of Harmonic Univalent Maps. Pacific Journal of Mathematics, 141, 1-20.
http://dx.doi.org/10.2140/pjm.1990.141.1 [9] Clunie, J. and Sheil-Smal, T. (1984) Harmonic Univalent Functions. Annales Academic Scientiarum Fennice Mathematica, 9, 3-25. [10] Duren, P. and Schober, G. (1987) A Variational Method for Harmonic Mappings on Convex Regions. Complex Variables, Theory and Application, 9, 153-168.
http://dx.doi.org/10.1080/17476938708814259 [11] Duren, P. and Schober, G. (1989) Linear Extremal Problems for Harmonic Mappings of the Disk. Proceedings of the American Mathematical Society, 106, 967-973.
http://dx.doi.org/10.1090/S0002-9939-1989-0953740-5 [12] Hengartner, W. and Schober, G. (1986) On the Boundary Behavior of Orientation-Preserving Harmonic Mappings. Complex Variables, Theory and Application, 5, 197-208.
http://dx.doi.org/10.1080/17476938608814140 [13] Hengartner, W. and Schober, G. (1986) Harmonic Mappings with Given Dilatation. Journal London Mathematical Society, 33, 473-483.
http://dx.doi.org/10.1112/jlms/s2-33.3.473 [14] Jun, S.H. (1993) Univalent Harmonic Mappings on Proceedings of the American Mathematical Society, 119, 109-114.
http://dx.doi.org/10.1090/S0002-9939-1993-1148026-3 [15] Nitsche, J.C.C. (1989) Lectures on Minimal Surfaces. Vol. 1, Translated from the German by Jerry M. Feinberg, Cambridge University Press, Cambridge. [16] Osserman, R. (1986) A Survey of Minimal Surfaces. 2nd Edition, Dover, New York. [17] Baernstein, A. (1974) Integral Means, Univalent Functions and Circular Symmetrization. Acta Mathematica, 133, 139169.
http://dx.doi.org/10.1007/BF02392144 [18] Duren, P. (1983) Univalent Functions. Springer-Verlag, Berlin.