2017/07/03 by Ali, Rosihan M., Lee, See Keong, Mondal, Saiful R.
#30C45 #33C10 #Complex Variables (math.CV) #FOS: Mathematics
paper · doi:10.48550/arxiv.1707.00379
For a fixed a ∈ \1, 2, 3, …\, the radius of starlikeness of positive order is obtained for each of the normalized analytic functions \mathttfa, ν(z)amp;:= (2a ν-a+1 a-(a(aν-a+1))/(2) Γ(a ν+1) a\mathttB2a-1, a ν-a+1, 1(aa/2 z))^\tfrac1a ν-a+1,
\mathttga, ν(z)amp;:= 2a ν-a+1 a-(a)/(2)(aν-a+1) Γ(a ν+1) za-aν a\mathttB2a-1, a ν-a+1, 1(aa/2 z),
\mathttha, ν(z)amp;:= 2a ν-a+1 a-(a)/(2)(aν-a+1) Γ(a ν+1) z(1)/(2)(1+a-aν) a\mathttB2a-1, a ν-a+1, 1(aa/2 √(z)) in the unit disk, where a\mathttBb, p, c is the generalized Bessel function a\mathttBb, p, c(z):= ∑k=0^∞ \frac(-c)kk! \mathrmΓ( a k +p+(b+1)/(2)) ((z)/(2))2k+p. The best range on ν is also obtained for a fixed a to ensure the functions \mathttfa, ν and \mathttga, ν are starlike of positive order in the unit disk. When a=1, the results obtained reduced to earlier known results.