On Subclass of Bazilevi·c Function B1(®), It's Distortion and the Fekete-SzegÄo Problem

Marjono Marjono

Abstract


In this paper we present the distortion and the Fekete-SzegÄo problem of subclass of Bazilevi·c functions, B1(®). First, we present the result of Singh concerning the sharp value of the coe±cients for B1(®), ja2j, ja3j and ja4j. Second, we give a solution of the Fekete-SzegÄo problem, i.e. an estimate
of ja3 ¡ ¹a 2 2j for any real and complex numbers ¹ where a2 and a3 are the coe±cients of functions f in B1(®), where B1(®) is de¯ned by (2), i.e. for each ® > 0 and for z 2 D, Re f 0 (z) f(z)
z ®¡1 > 0. These results are sharp for the functions f0 de¯ned by (3) for any real number ¹ which satis¯es ¹ < (1 ¡ ®)=2, or ¹ ¸ (4 + 3® + ® )=[2(2 + ®)] and for any complex number ¹ which satis¯es j3 + ® ¡ 2¹(2 + ®)j ¸ (1 + ®) 2 . These results are sharp for the functions f1 de¯ned by (4) for the other real and complex numbers ¹. Next, we use similar methods to get estimates for linear expressions involving higher coe±cients of function in B1(®). 2

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References


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DOI: http://dx.doi.org/10.12962/j1829605X.v3i1.1391

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Limits: Journal Mathematics and its Aplications by Pusat Publikasi Ilmiah LPPM Institut Teknologi Sepuluh Nopember is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.
Based on a work at https://iptek.its.ac.id/index.php/limits.