The Estimations Of The Upper Bound Of The Norm Of Schwarz Derivative Of Alexander Transformation Of Two Classes Of Spiral Functions

In this paper,we estimate the upper bound of the norm of Schwarzian derivatives of the Alexander trans;forms of α-spiral-like functions and α-spiral-like functions of order r based on the properties of function with positive real part and obtain three main theorems and one corollary.we use the following formula to express the Alexander transforms of g（z）where1.Let C_be the class of the Alexander transforms of F_α which areα-spiral-like functions,i.e.We estimate the upper bound of the norm of Schwarzian derivatives of the Alexander transforms of F_α and obtain the following theorem and corollary.Theorem 1Corollaryl2.Let C_α,r be the class of the Alexander transforms of F_α,r which areα-spiral-like functions of order r,i.e.We estimate the upper bound of the norm of Schwarzian derivatives of the Alexander transforms of α-spiral-like functions of order r and obtain thefollowing theorems.Theorem 2 Suppose that f（z）∈ C_α,r,thenTheorem 3A.1).If α∈（-0.2269,0.2269）,i.e.α∈（-13°,13°）,there are r₁（α）,r2（α）∈（0,0.375）and r₁（α）<r2（α）,such that① for any r∈（0,r₁（α））,G（α,r）is decreasing function with r;② for any r ∈（r₁（α）,r2（α））,G（α,r）is increasing function with r;⑧ for any（?）r∈（r2（α）,1）,G（α,r）is decreasing function with r ·i.e,α∈（-90°,13°）∪（13°,90°）.For r ∈（0,1）,then G（α,r）is decreasing function with r.B.There is a constant r0 ∈（0.64,0.65）,such that1).For any r∈（r0,1）,then G（α,r）is increasing function with...