Abstract
We derive some results for a new class of analytic functions defined by using Salagean operator. We give some properties of functions in this class and obtain numerous sharp results including for example, coefficient estimates, distortion theorem, radii of star-likeness, convexity, close-to-convexity, extreme points, integral means inequalities, and partial sums of functions belonging to this class. Finally, we give an application involving certain fractional calculus operators that are also considered.
Highlights
Let AA denote the class of functions of the form ff = zz z aakkzzkk kkkk that are analytic and univalent in the open unit disc UU U UUU U C ∶ |zzz z zz
For two functions ff and gg, analytic in UU, we say that the function ffffff is subordinate to gggggg in UU and write ffffff f ffffff, if there exists a Schwarz function wwwwww, which is analytic in UU with wwwwwww and |wwwwwwwww, such that fffffff ggggggggggggggggg
It is known that ff ≺ gg ⟹ ff (0) = gg (0), ff (UU) ⊂ gg (UU) . (3)
Summary
Kkkk that are analytic and univalent in the open unit disc UU U UUU U C ∶ |zzz z zz. For fffffffff, Salagean [1] introduced the following differential operator: DD0fffffff ffffff, DD1fffffffffff′(zzzz z, DDnnffffffffffffnnnnfffffffffff ff fffffffff. If the function gg is univalent in UU, we have the following equivalence [2, page 4]: ff (zz) ≺ gg (zz) ⟺ ff (0) = gg (0) , ff (UU) ⊂ gg (UU) . Let UUmmmmm(βββ βββ βββ denote the subclass of AA consisting of functions ffffff of the form (1) and satisfy the following subordination: DDmmff (zz) DDnnff (zz). (see Shams et al [5, 6]); (iii) UU1,0 (0, AAA AA) = SS∗(AAA AA) = ff f ff f zzzz′ (zz) ≺ 1 + AAAA ff (zz) 1 + BBBB (−1 ≤ BBBBBBBBBBBBB) , (8). Let TT denote the subclass of functions of AA of the form ff (zz) = zz z aakkzzkk, aakk ≥ 0. (iii) TTTT1,0(0,1−2ααααααααα∗(ααα and TTTT2,1(0,1− 2ααααααα ααααααααααα α αα (see Silverman [12])
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