Abstract
We utilize a fractional calculus operator and we introduce a new subclass of analytic functions. We
investigate the characteristics properties of the derived new subclass and we explore relevant topics.
Results & Lemmas (6)
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Theorem 1.
Theorem 1. A function 𝑓(𝑡) ∈𝐴(𝛼, 𝛽, 𝛾) if and only if 𝑓(𝑡) can be expressed as 𝑓(𝑡) = 𝑡+ 2(𝛼+ 𝛽−𝛾) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) ∫ (∑𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+…
Theorem 1. A function 𝑓(𝑡) ∈𝐴(𝛼, 𝛽, 𝛾) if and only if 𝑓(𝑡) can be expressed as 𝑓(𝑡) = 𝑡+ 2(𝛼+ 𝛽−𝛾) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) ∫ (∑𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1)𝑥𝑛−1 (𝛤(𝑛+ 1)) 2(𝛼+ 𝑛𝛽) ∞ 𝑛=2 𝑧𝑛) |𝑥|=1 𝑑𝜇(𝑥) (1) Where 𝜇(𝑥)is the probability measure on 𝑋= {𝑥: |𝑥| = 1}. For fixed 𝛼, 𝛽 And 𝛾, 𝐴(𝛼, 𝛽, 𝛾)and the probability measures {𝜇} defined on 𝑋 Are one-to-one by the assertion (1)
Corollary 1.
Corollary 1. The extreme points of the class 𝐴(𝛼, 𝛽, 𝛾) are 𝑓𝑥(𝑡) = 𝑧+ 2(𝛼+ 𝛽−𝛾) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) ∑𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1)𝑥𝑛−1 (𝛤(𝑛+ 1))…
Corollary 1. The extreme points of the class 𝐴(𝛼, 𝛽, 𝛾) are 𝑓𝑥(𝑡) = 𝑧+ 2(𝛼+ 𝛽−𝛾) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) ∑𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1)𝑥𝑛−1 (𝛤(𝑛+ 1)) 2(𝛼+ 𝑛𝛽) ∞ 𝑛=2 𝑡𝑛, (|𝑥| = 1)
Corollary 2.
Corollary 2. If 𝑓(𝑡) = 𝑡+ ∑ 𝑎𝑛 ∞ 𝑛=2 𝑡𝑛∈𝐴(𝛼, 𝛽, 𝛾), then for 𝑛≥2,we have |𝑎𝑛| ≤2(𝛼+ 𝛽−𝛾)𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆)(𝛤(𝑛+ 1))…
Corollary 2. If 𝑓(𝑡) = 𝑡+ ∑ 𝑎𝑛 ∞ 𝑛=2 𝑡𝑛∈𝐴(𝛼, 𝛽, 𝛾), then for 𝑛≥2,we have |𝑎𝑛| ≤2(𝛼+ 𝛽−𝛾)𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆)(𝛤(𝑛+ 1)) 2(𝛼+ 𝑛𝛽)
Corollary 3.
Corollary 3. If 𝑓(𝑡) = 𝑡+ ∑ 𝑎𝑛 ∞ 𝑛=2 𝑧𝑛∈𝐴(𝛼, 𝛽, 𝛾), the for |𝑡| = 𝑟< 1,we have |𝑓(𝑡)| ≤𝑟+ 2(𝛼+ 𝛽−𝛾) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) ∑𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+…
Corollary 3. If 𝑓(𝑡) = 𝑡+ ∑ 𝑎𝑛 ∞ 𝑛=2 𝑧𝑛∈𝐴(𝛼, 𝛽, 𝛾), the for |𝑡| = 𝑟< 1,we have |𝑓(𝑡)| ≤𝑟+ 2(𝛼+ 𝛽−𝛾) 𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) ∑𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1)𝑟𝑛 (𝛤(𝑛+ 1)) 2(𝛼+ 𝑛𝛽) ∞ 𝑛=2
Theorem 2.
Theorem 2. Let 𝑓(𝑡) ∈𝐴(𝛼, 𝛽, 𝛾),then 𝑓(𝑡)is univalent in |𝑡| < 𝑅(𝛼, 𝛽, 𝛾),where 𝑅(𝛼, 𝛽, 𝛾) = Inf 𝑛 (𝑛𝛽+ 𝛼)(𝛤(𝑛+ 1)) 2𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆)…
Theorem 2. Let 𝑓(𝑡) ∈𝐴(𝛼, 𝛽, 𝛾),then 𝑓(𝑡)is univalent in |𝑡| < 𝑅(𝛼, 𝛽, 𝛾),where 𝑅(𝛼, 𝛽, 𝛾) = Inf 𝑛{ (𝑛𝛽+ 𝛼)(𝛤(𝑛+ 1)) 2𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆) 2𝑛(𝛼+ 𝛽−𝛾)𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1)} 1 𝑛−1 . This result is sharp.
Theorem 3.
Theorem 3. If 𝑓(𝑡) ∈𝐴(𝛼, 𝛽, 𝛾) then 𝑓(𝑡) is starlike of order 𝜇 |𝑡| < 𝑟0, 0 ≤𝜇< 1, Where 𝑟0 = Inf 𝑛 (1 −𝜇)𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆)(𝛤(𝑛+ 1)) 2(𝑛𝛽+…
Theorem 3. If 𝑓(𝑡) ∈𝐴(𝛼, 𝛽, 𝛾) then 𝑓(𝑡) is starlike of order 𝜇 |𝑡| < 𝑟0 , 0 ≤𝜇< 1, Where 𝑟0 = Inf 𝑛{ (1 −𝜇)𝛤(2 −𝜂)𝛤(2 + 𝜂−𝜆)(𝛤(𝑛+ 1)) 2(𝑛𝛽+ 𝛼) 2(𝑛−𝜇)(𝛼+ 𝛽−𝛾)𝛤(𝑛+ 𝜂−𝜆+ 1)𝛤(𝑛−𝜂+ 1)} 1 𝑛−1
Function classes studied:
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