Singular values of two-parameter families @l((b^z-1)/z)^@m and @l(z/(b^z-1))^@h

摘要

The singular values of two kinds of two-parameter families of functions (i) f"@l","@m(z)=@l((b^z-1)/z)^@m and f"@l","@m(0)=@l(lnb)^@m, @m>0, (ii) g"@l","@h(z)=@l(z/(b^z-1))^@h and g"@l","@h(0)=@l/(lnb)^@h, @h>0; @l@?@?@?{0}, z@?@?, b>0, b 1 are described. It is shown that all the critical values of f"@l","@m(z) and g"@l","@h(z) lie interior and exterior of the disk centered at origin and having radii |@l(lnb)^@m| and |@l/(lnb)^@h| respectively. Further, it is proved that both the functions f"@l","@m(z) and g"@l","@h(z) have infinitely many singular values for all b>0, b 1.