@article{Bukhari2023, 
author = {Syed Zakar Hussain Bukhari and Abbas Kareem Wanas and Mohamed Abdalla and Sidra Zafar},
title = {Region of variablity for Bazilevic functions},
year = {2023},
journal = {AIMS Mathematics},
volume = {8},
number = {11},
pages = {25511-25527},
keywords = {spirallike and Bazilevičfunctions, Schwarz lemma},
url = {https://www.sciopen.com/article/10.3934/math.20231302},
doi = {10.3934/math.20231302},
abstract = {Let        H   be the family of analytic functions defined in an open unit disk        U    =        {          z      :              |            z              |            <      1        }   and         A    =      {          f      ∈              H            :      f      (      0      )      =              f                                                        ′                                          (      0      )      −      1      =      0      ,                                                          (      z      ∈              U            )        }    .For    A  ∈      C    ,  B  ∈  [  −  1  ,  0  ) and    γ  ∈      (                            −          π                2            ,              π        2              )    , a function    h  ∈              P              γ        [  ξ  ,  A  ;  B  ] can be written as:     h  (  z  )  =  cos  ⁡  γ            1      +      A      ω      (      z      )              1      +      B      ω      (      z      )        +  i  sin  ⁡  γ  ,    (  ω  (  0  )  =  0  ,      |          ω      (      z      )        |    <  1  ,  z  ∈      U    )  ,where    ξ  =      ω          ′            (    0    )    ∈            U        ¯  . The family              B              γ            [          ψ      ,      ξ      ,      β      ,      A      ;      B        ]   contains analytic functions    f in        U   such that                       e                  i          γ                    z              f                  ′                    (      z      )                                [                      f                          (              z              )                                ]                          1          −          β                                      [                      ψ            (            z            )                    ]                          β                      ∈            P              γ        [  ξ  ,  A  ;  B  ]  ,where    ψ is a starlike function. In this research, we find the region of variability denoted by              V              γ        [  ψ  ,      z          0        ,  ξ  ,  A  ;  B  ] for    f      (          z              0              )  , where    f is ranging over the family              B              γ            [          ψ      ,      ξ      ,      β      ,      A      ;      B        ]   for any fixed        z          0        ∈      U   and    ξ  ∈            U        ¯    .}
}