Using Jackson's $ q $-derivative operator, we introduce $ q $-analogues of the function classes $ \mathcal{QCV}(\mu, A, B) $ and $ \mathcal{QST}(\eta, A, B) $ of Altintaş and Kılıç, consisting of analytic functions in the open unit disk $ \mathbb{D} $. The subordination conditions use the $ q $-dependent Janowski function $ (1+A_q z)/(1+B_q z) $, where $ A_q = \frac{(A+1)+(A-1)q}{2} $ and $ B_q = \frac{(B+1)+(B-1)q}{2} $ are affine in $ q $, remain in the admissible range $ -1 < B_q < A_q\le1 $ for every $ q\in(0, 1) $, tend to $ A, B $ as $ q\to1^- $, and for $ (A, B) = (1, -1) $ give $ (1+z)/(1-qz) $, the function of the $ q $-starlike class of Ismail, Merkes, and Styer. We derive the $ q $-Alexander relation $ D_q(zD_q f) = D_q f+qzD_q^2f $, prove coefficient bounds for the auxiliary $ q $-starlike functions, and obtain, with the $ q $-integer arithmetic carried out carefully, upper bounds for the coefficients $ a_n $ of both new classes in terms of $ q $-integers. Several known subclasses appear as special cases, and the classical estimates are obtained in the limit $ q\to1^- $.
Citation: Shujaat Ali Shah, Summaiya Ayoub Shaikh, Dalal Alhwikem, Sheza M. El-Deeb. Coefficient estimates for $ q $-analogues of quasi-convex function classes via generalized $ q $-Janowski parameters[J]. AIMS Mathematics, 2026, 11(11): 36233-36243. doi: 10.3934/math.20261403
Using Jackson's $ q $-derivative operator, we introduce $ q $-analogues of the function classes $ \mathcal{QCV}(\mu, A, B) $ and $ \mathcal{QST}(\eta, A, B) $ of Altintaş and Kılıç, consisting of analytic functions in the open unit disk $ \mathbb{D} $. The subordination conditions use the $ q $-dependent Janowski function $ (1+A_q z)/(1+B_q z) $, where $ A_q = \frac{(A+1)+(A-1)q}{2} $ and $ B_q = \frac{(B+1)+(B-1)q}{2} $ are affine in $ q $, remain in the admissible range $ -1 < B_q < A_q\le1 $ for every $ q\in(0, 1) $, tend to $ A, B $ as $ q\to1^- $, and for $ (A, B) = (1, -1) $ give $ (1+z)/(1-qz) $, the function of the $ q $-starlike class of Ismail, Merkes, and Styer. We derive the $ q $-Alexander relation $ D_q(zD_q f) = D_q f+qzD_q^2f $, prove coefficient bounds for the auxiliary $ q $-starlike functions, and obtain, with the $ q $-integer arithmetic carried out carefully, upper bounds for the coefficients $ a_n $ of both new classes in terms of $ q $-integers. Several known subclasses appear as special cases, and the classical estimates are obtained in the limit $ q\to1^- $.
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Shujaat Ali Shah, Summaiya Ayoub Shaikh, Dalal Alhwikem, Sheza M. El-Deeb. Coefficient estimates for $ q $-analogues of quasi-convex function classes via generalized $ q $-Janowski parameters[J]. AIMS Mathematics, 2026, 11(11): 36233-36243. doi: 10.3934/math.20261403
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