@article{Khan2026, 
author = {Bilal Khan and Fairouz Tchier and Manuela Oliveira},
title = {Regions of variability for generalized Janowski functions},
year = {2026},
journal = {AIMS Mathematics},
volume = {11},
number = {2},
pages = {3499-3511},
keywords = {region of variability, Janowski functions, generalized Janowski functions, Schwarz function},
url = {https://www.sciopen.com/article/10.3934/math.2026142},
doi = {10.3934/math.2026142},
abstract = {Let    r  ∈      C    ,    s  ∈  [  −  1  ,  0  ),    0  ≤  α  <  1. Then,        Q        [          r      ,      s      ,      α        ]   stands for the set of analytic functions    q that is within the open unit disk    E, with    q      (    0    )    =  1  , and satisfies the explicit representation     q      (    ζ    )    =            1      +              (                              (                          1              −              α                        )                    r          +          α          s                )            χ              (        ζ        )                    1      +      s      χ              (        ζ        )              ,where    χ      (    0    )    =  0 and $ \left \vert \chi \left(\zeta \right)\right \vert < 1.    I  n  t  h  i  s  a  r  t  i  c  l  e  ,  w  e  f  i  n  d  t  h  e  r  e  g  i  o  n  s  o  f  v  a  r  i  a  b  i  l  i  t  y W_{\lambda }\left(\zeta _{0}, r, s, \alpha \right)     f  o  r \int \limits_{0}^{z_{0}}q\left(\rho \right) d\rho \     w  h  e  n q    r  a  n  g  e  s  o  v  e  r  t  h  e  c  l  a  s  s \mathcal{Q}_{\lambda }\left[r, s, \alpha \right]  $ defined as               Q              λ            [          r      ,      s      ,      α        ]    =      {          q      ∈              Q                    [                  r          ,          s          ,          α                ]            :              q                  ′                            (        0        )            =              (                              (                          1              −              α                        )                                (                          r              −              s                        )                          )            λ        }  for any fixed        ζ          0        ∈  E and    λ  ∈      E    ¯  . As a corollary, the region of variability appears for the alternate sets of parameters as well.}
}