Sufficient Conditions for Starlikeness

We obtain the conditions on <TEX>β</TEX> so that <TEX>1+βzp^(z)1+4z/3+2z²/3</TEX> implies p(z) <TEX></TEX> (2+z)/(2-z), <TEX>1+(1-α)z</TEX>, <TEX>(1+(1-2α)z)/(1-z)</TEX>, (<TEX>0≤α</TEX><1), exp(z) or <TEX>√1+z</TEX>. Similar results are obtained by considering the expressions <TEX>1+βzp^(z)/p(z)</TEX>, <TEX>1+βzp^(z)/p²(z)</TEX> and <TEX>p(z)+βzp^(z)/p(z)</TEX>. These results are applied to obtain sufficient conditions for normalized analytic function f to belong to various subclasses of starlike functions, or to satisfy the condition <TEX>log(zf^(z)/f(z))</TEX> < 1 or <TEX>(zf^(z)/f(z))²-1</TEX> < 1 or zf'(z)/f(z) lying in the region bounded by the cardioid <TEX>(9x²+9y²-18x+5)²-16(9x²+9y²-6x+1)=0</TEX>.