276,397 research outputs found
Sufficient Conditions for Starlike Functions Associated with the Lemniscate of Bernoulli
Let -1\leq B<A\leq 1. Condition on \beta, is determined so that 1+\beta
zp'(z)/p^k(z)\prec(1+Az)/(1+Bz)\;(-1<k\leq3) implies p(z)\prec \sqrt{1+z}.
Similarly, condition on \beta is determined so that 1+\beta zp'(z)/p^n(z) or
p(z)+\beta zp'(z)/p^n(z)\prec\sqrt{1+z}\;(n=0, 1, 2) implies
p(z)\prec(1+Az)/(1+Bz) or \sqrt{1+z}. In addition to that condition on \beta is
derived so that p(z)\prec(1+Az)/(1+Bz) when p(z)+\beta
zp'(z)/p(z)\prec\sqrt{1+z}. Few more problems of the similar flavor are also
considered
Two Results on Homogeneous Hessian Nilpotent Polynomials
Let and the Laplace operator. A formal power series is said to be {\it
Hessian Nilpotent}(HN) if its Hessian matrix \Hes P(z)=(\frac {\partial^2
P}{\partial z_i\partial z_j}) is nilpotent. In recent developments in [BE1],
[M] and [Z], the Jacobian conjecture has been reduced to the following
so-called {\it vanishing conjecture}(VC) of HN polynomials: {\it for any
homogeneous HN polynomial of degree , we have for any .} In this paper, we first show that, the VC holds
for any homogeneous HN polynomial provided that the projective
subvarieties and of determined by the principal ideals generated by and
, respectively, intersect only at regular
points of . Consequently, the Jacobian conjecture holds for the
symmetric polynomial maps with HN if has no non-zero
fixed point with . Secondly, we show
that the VC holds for a HN formal power series if and only if, for any
polynomial , when .Comment: Latex, 7 page
- β¦