#### Glasnik Matematicki, Vol. 39, No.2 (2004), 265-272.

### ON THE LINEAR COMBINATION OF THE
REPRESENTATIONS OF STARLIKENESS AND CONVEXITY

### Nikola Tuneski and Roza Aceska

Faculty of Mechanical Engineering, Karpos II b.b.,
1000 Skopje, Republic of Macedonia

*e-mail:* `nikolat@mf.ukim.edu.mk`

*e-mail:* `aroza@mf.ukim.edu.mk`

**Abstract.** Let
A be the class of
analytic functions in the unit disk
*U* = {*z* : |*z*| < 1}
that are normalized with *f*(0) = *f*'(0) - 1 = 0.
Also, let *S**[*A*,*B*],
-1 ≤ *B* < *A* ≤ 1, be the class of functions
*f* ∈
A,
such that
*z**f*'(*z*) / *f*(*z*)
(1 + *Az*) / (1 + *Bz*),
where ""
denotes the usual subordination. In this paper we
investigate the linear combination of the analytic representations
of starlikeness and convexity and give sharp sufficient
conditions over the differential operator

*a* *z**f*'(*z*) / *f*(*z*) + *b* (1 + *z**f*''(*z*) /
*f*'(*z*))

that imply *f* ∈
*S**[*A*,*B*].
In that purpose we use the method of
differential subordinations. Several corollaries and examples for
different choices of *A*, *B*, *a* and *b*
are given and comparison with previous known results is done.
**2000 Mathematics Subject Classification.**
30C45.

**Key words and phrases.** Starlike function, starlike
function of order α, criteria, differential subordination,
Jack lemma.

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