Mathematics & Statistics

Lemma 2.1. [20] For 0 ⩽α ⩽β, we have f′ ∈K(α, β) if and only if there exists g ∈ST((2 −β + α)/2) and ϕ ∈R such that arg eiϕ zf′(z) g(z) ⩽απ 2 , z ∈D. (2.2) Define the function p by p(z) = eiϕzf′(z)/g(z) where f′ ∈K(α, β) and g ∈ST((2 −β + α)/2). We note that zp′(z) p(z) = 1 + zf′′(z) f′(z) −zg′(z) g(z) . (2.3)

Theorem 3.1. Let 0 ⩽α ⩽β. The sharp CVNe radius for functions whose derivative belongs to K(α, β) is given by RCVNe := 4 3(α + β) + p 9(α + β)2 + 8(3(β −α) + 2).

Corollary 3.2. Radius of convexity associated with the class CVNe for some special cases: (1) The CVNe radius for the class CC(σ, 0) is RCVNe = 4 3(2 −σ) + √ 9σ2 −38σ + 54. (2) The CVNe radius for the class Vk is RCVNe = 4 3k + √ 9k2 + 64. (3) The CVNe radius for the class SCC(α) is RCVNe =

Theorem 3.3. Let 0 ⩽α ⩽β. The CVh radius for functions whose derivative belongs to K(α, β) is given by RCVh = 2 sinh−1(1) (α + β) + q (α + β)2 + 4(β −α + sinh−1(1))(sinh−1(1)) . The radius obtained is sharp.

Corollary 3.4. Radius of convexity associated with the class CVh for some special cases: (1) The CVh radius for the class CC(σ, 0) is RCVh = sinh−1(1) (2 −σ) + q (2 −σ)2 + (2(1 −σ) + sinh−1(1))(sinh−1(1)) . (2) The CVh radius for the class Vk is RCVh = 2 sinh−1(1) k + q k2 + 4(sinh−1(1))(2 + sinh−1(1)) .

Theorem 3.5. Let 0 ⩽α ⩽β. The sharp CVSG radius for functions f such that f′ ∈ K(α, β) is given by RCVSG = 2(e −1) (α + β)(e + 1) + p (e + 1)2(α + β)2 + 4(e −1)((e + 1)(β −α) + e −1).

Corollary 3.6. Radius of convexity associated with the class CVSG for some special cases: (1) The CVSG radius for the class CC(σ, 0) is RCVSG = (e −1) (2 −σ)(e + 1) + p (e + 1)2(2 −σ)2 + ((3e + 1) −2σ(e + 1))(e −1). (2) The CVSG radius for the class Vk RCVSG = 2(e −1) k(e + 1) + p (e + 1)2k2 + 4(3e + 1)(e −1). (3) The CVSG radius for the class SCC(α) is RCVSG =

Theorem 3.7. Let 0 ⩽α ⩽β. The sharp CVsin radius for functions f such that f′ ∈ K(α, β) is given by RCVsin = 2(sin 1) (α + β) + p (α + β)2 + 4(β −α + sin 1)(sin 1).

Corollary 3.8. Radius of convexity associated with the class CVsin for some special cases: (1) The CVsin radius for the class CC(σ, 0) is RCVsin = sin 1 (2 −σ) + p (2 −σ)2 + (2 −2σ + sin 1)(sin 1). (2) The CVsin radius for the class Vk is RCVsin = 2 sin 1 k + p k2 + 4(2 + sin 1)(sin 1).

Theorem 3.9. The radius of lemniscate convexity for the functions whose derivative belongs to K(α, β), 0 ⩽α ⩽β is given by RCVL = 2( √ 2 −1) (α + β) + q (α + β)2 + 4( √ 2 −1)( √ 2 + (β −α −1)) .

Corollary 3.10. Radius of convexity associated with the class CVL for some special cases:

Theorem 3.11. Let 0 ⩽α ⩽β. The sharp CVR radius for functions f such that f′ ∈ K(α, β) is given by RCVR = ( R2, R2 ⩽R1 R3, R2 ⩾R1 where R1 =

Corollary 3.12. Radius of convexity associated with the class CVR for some special cases: (1) The CVR radius for the class CC(σ, 0) is RCVR = 3 −2 √ 2 2(2 −σ) + q 4((2 −σ)2 + (6 −4 √ 2)σ −8 √ 2 + 11) .

Theorem 3.13. Let 0 ⩽α ⩽β. The sharp CV$ radius for functions f such that f′ ∈ K(α, β) is given by RCV$ = ( R2, R2 ⩽R1 R3, R2 ⩾R1 where R1 =

Corollary 3.14. Radius of convexity associated with the class CV$ for some special cases: (1) The CV$ radius for the class CC(σ, 0) is RCV$ = (2 − √ 2) (2 −σ) + q (2 −σ)2 + ( √ 2 −2σ)( √ 2 −2) .$

Theorem 3.15. Let 0 ⩽α ⩽β. The sharp CVC radius for functions f such that f′ ∈ K(α, β) is given by RCVC = ( R2, R2 ⩽R1 R3, R2 ⩾R1 where R1 =  2 3(β −α) + 2 1/2 ,

Corollary 3.16. Radius of convexity associated with the class CVC for some special cases: (1) The CVC radius for the class CC(σ, 0) is RCVC = 2 3(2 −σ) + √ 9σ2 −24σ + 28. (2) The CVC radius for the class Vk is RCVC = 4 3k + √ 9k2 −32.

Theorem 3.17. Let 0 ⩽α ⩽β. The sharp CVe radius for functions f such that f′ ∈ K(α, β) is given by RCVe = ( R2, R2 ⩽R1 R3, R2 ⩾R1 where R1 =

Corollary 3.18. Radius of convexity associated with the class CVe for some special cases: (1) The CVe radius for the class CC(σ, 0) is RCVe = e −1 e(2 −σ) + p e2(2 −σ)2 −(e(1 −2σ) + 1)(e −1). (2) The CVe radius for the class Vk is RCVe = 2(e −1) ek + p e2(k2 −4) + 4.

Theorem 3.19. Let 0 < γ ⩽1 and 0 ⩽α ⩽β. The radius of uniform convexity for functions whose derivative belongs to K(α, β) is given by RUCV = R2 if R2 ⩽R1 and RUCV ⩾R3, if R2 ⩾R1 where R1 =  1 2(β −α) + 1 1/2 , R2 = 1 (α + β) + p (α + β)2 −(2(β −α) −1) and

Theorem 3.20. Let 0 < γ ⩽1 and 0 ⩽α ⩽β. The radius of strong convexity of order γ for functions whose derivative belongs to K(α, β) is given by RSCV(γ) ⩾ 2 sin πγ 2 (α + β) + q (α + β)2 −4(β −α −1)(sin πγ 2 )2 . Acknowledgements The authors thank the referees for their comments and valuable suggestions. Author contributions. All the co-authors have contributed equally in all aspects of the preparation of this submission. Conflict of interest statement. The authors declare that they have no known