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Information Center for Mathematical Science

### 논문검색

Information Center for Mathematical Science

#### 논문검색

Glasnik Matematicki Serja III.
( Vol. 39 NO.2 / (2004))
On the linear combination of the representations of starlikeness and convexity

Pages. 265-272
Abstract Let ${cal A}$ be the class of analytic functions in the unit disk ${cal U} = {{it z} : |{it z}| < 1}$ that are normalized with ${it f}(0) = {it f}'(0) - 1 =0$. Also, let ${sl S}^*[{sl A,B}], -1 leq {sl B} < {sl A} leq 1$, be the class of functions ${it f} in {cal A}$, such that $frac{{it zf}'({it z})}{{it f}({it z})} prec frac{1+{sl A}{it z}}{1+{sl B}{it z}}$, where $"prec"$ 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 $${it a}frac{{it zf}'({it z})}{{it f}({it z})} + {it b}(1+frac{{it zf}"({it z})}{{it f}'({it z})})$$ that imply ${it f} in {sl S}^*[{sl A,B}]$. In that purpose we use the method of differential subordinations. Several corollaries and examples for different choices of ${sl A, B}, {it a}$ and ${it b}$ are given and comparison with previous known results is done. 1.Introduction and preliminaries 2.Main results and consequences Starlike function, starlike function of order $\alpha$, criteria, differential subordination, Jack lemma 30C45