Bohr phenomenon for certain close-to-convex analytic functions

Lemma 1.7. [23] Let f ∈S∗(φ). Then zf ′(z)/f(z) ≺zh′(z)/h(z) and f(z)/z ≺ h(z)/z.

Lemma 1.8. [23] Let f ∈C(φ). Then zf ′′(z)/f ′(z) ≺zk′′(z)/k′(z) and f ′(z) ≺ k′(z). Ma-Minda functions φ have been considered with the condition φ′(0) > 0. Mo- tivated by this, recently, Kumar and Banga [22] have introduced the function Φ, called non-Ma-Minda function, with the condition Φ′(0) < 0 and the other condi- tions on Φ are same as that of φ. Note that Φ can obtained from φ by a rotation, namely, z by −z. By going a similar manner as the definition of S∗(φ) and C(φ) (see [23]), Kumar an

Lemma 1.9. [26] Let min|z|=r |φ(z)| = φ(−r), max|z|=r |φ(z)| = φ(r), |z| = r. If f ∈Cs(φ), then 1 r r Z 0 φ(−r)(k′(−r2))1/2 dr ≤|f ′(z)| ≤1 r r Z 0 φ(r)(k′(r2))1/2 dr. From [30, Theorem 9], for f ∈Cs(φ), we have (1.10)

Lemma 1.12. [17] Let f(z) = z + al+1zl+1 + · · · ∈C(φ), then we have (k′(−rl))1/l ≤|f ′(z)| ≤(k′(rl))1/l. The bounds are sharp for some suitable rotations of the function Kl which is defined by Kl(z) = z Z 0 (k′(ξl))1/l dξ, z ∈D, where k is defined in (1.6). In particular for l = 2 we can obtain the bounds of |f ′(z)| for odd convex functions. From Lemma 1.12, the following can be easily obtained for l = 2 r Z

Lemma 1.14. [26] Let min|z|=r |φ(z)| = φ(−r), max|z|=r |φ(z)| = φ(r), |z| = r. If f ∈S∗ c (φ), then (i) h′(−r) ≤|f ′(z)| ≤h′(r)

Lemma 1.15. [26] Let min|z|=r |φ(z)| = φ(−r), max|z|=r |φ(z)| = φ(r), |z| = r. If f ∈Cc(φ), then (i) k′(−r) ≤|f ′(z)| ≤k′(r) (ii) −k(−r) ≤|f(z)| ≤k(r) (iii) f(D) ⊇{w : |w| ≤−k(−1)}. The results are sharp. Motivated by the class S∗ s, Gao and Zhou [16] have studied the class Ks of close- to-convex univalent functions, where Ks is the class of functions f ∈S satisfying the condition Re  z2f ′(z) g(z)g(−z)  < 0,

Lemma 1.17. [15] Let min|z|=r |φ(z)| = φ(−r), max|z|=r |φ(z)| = φ(r), |z| = r. If f ∈Ks(φ), then the following sharp inequalities hold: (i) φ(−r) 1 + r2 ≤|f ′(z)| ≤φ(r) 1 −r2 (|z| = r < 1) (ii) r Z 0 φ(−t) 1 + t2 dt ≤|f(z)| ≤ r Z 0 φ(t)

Lemma 1.19. [12] Let f and g be analytic in D with Taylor expansions (1.1) and (1.18) respectively and g ≺f, then (1.20) ∞ X n=0 |bn|rn ≤ ∞ X n=0 |an|rn for z| = r ≤1/3. In general, one obtains the Bohr radius for certain classes of analytic functions in D, when the sharp coefficient bounds for this class are known. But the sharp coefficient bounds for most of the Ma-Minda subclasses are not yet known. Using

Lemma 1.19, Allu and Halder [11] recently have obtained Bohr radius for certain classes of Ma-Minda starlike and convex functions. In this article, we consider cer- tain classes of close-to-convex functions associated with Ma-Minda functions e.g. S∗ c (φ), Cc(φ), Ks(φ) and Cs(φ). The sharp coefficient bounds of these classes are not yet known. Hence, we encounter the problem to find the best possible lower bound of the radius so that Bohr phenomenon holds for these classes. As a conse- quence, we a

Lemma 2.1. (i) Let f and g be analytic in D with series representation f(z) = P∞ n=1 anzn and (1.18) respectively such that f(z) = R z 0 g(ξ) dξ for z ∈D, where integration is taken along a linear segment joining 0 to z ∈D. Then Mf(r) = Z r 0 Mg(t) dt for |z| = r < 1. Here Mf(r) and Mg(r) are respectively the majorant series associated with f and g respectively.

Theorem 2.2. Let f ∈Ks(φ) be of the form (1.5). Then (2.3) |z| + ∞ X n=2 |an||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤Rf, where Rf = min{1/3, rf} and rf is the smallest positive root of R(r) = L(1) in (0, 1). Here R(r) := R r 0 (Mφ(t))/(1 −t2) dt , L(r) := R r 0 (φ(−t)) /(1+ t2) dt and Mφ is the associated majorant series of φ.

Theorem 2.2 are all positive i.e. Bn > 0 for n ≥1. Then the majorant series Mφ(r) = φ(r), 0 < r < 1 and hence R(r) := R r 0 (φ(t))/(1 −t2) dt. (ii) (Bohr phenomenon for the corresponding class Ks(Φ) associated with non- Ma-Minda functions) Let Φ be the corresponding non-Ma-Minda function of φ, which is actually a rotation by mere replacing z by −z. Therefore the image of the unit disk D under the functions Φ and φ are identical. Thus we conclude that Ks(Φ) = Ks(φ) and the Bohr phenomenon (2.3) h

Lemma 2.4. (Bohr phenomenon for the corresponding subordination class) Let q(z) = P∞ n=1 qnzn ∈SK f (φ) as defined in (1.16) and f be of the form (1.5). Then ∞ X n=1 |qn||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤Rf, where Rf is defined as in Theorem 2.2. For φ(z) = (1+(1−2γ)z)/(1−z), the class Ks(φ) reduces to Ks(γ). In particular, for γ = 0, Ks(φ) reduces to Ks.

Corollary 2.5. (i) (Bohr phenomenon for the class Ks(γ)) Any function f ∈Ks(γ) with 0 ≤γ < 0.259056404 satisfies the inequality (2.3) for |z| = r ≤rf, where rf is the root of (2.6) γ 2ln 1 + r 1 −r  + (1 −γ) r 1 −r = 1 −γ 2 ln2 + γπ

Corollary 2.7. The class Ks(α, β) satisfies the Bohr phenomenon (2.3) for |z| = r ≤Rf = min{1/3, rf}, where rf is the smallest root of (2.8) r Z 0 1 + βt (1 −αβt)(1 −t2) dt = 1 Z 0 1 −βt (1 + αβt)(1 + t2) dt in (0, 1).

Theorem 2.9. Let f ∈S∗ c (φ) be of the form (1.5). Then (2.10) |z| + ∞ X n=2 |an||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{1/3, rf} and rf is the smallest positive root of P(r) + h(−1) = 0 in (0, 1), where P(r) := R r 0 ((Mh(t)Mφ(t)) /t) dt. Here Mh(t) and Mφ(t) are the majorant series of h and φ respectively.

Lemma 2.12. Bohr phenomenon for the corresponding subordination class S∗ cf(φ)
Let g ∈S∗ cf(φ) be of the form g(z) = P∞ n=1 gnzn. Then (2.13) ∞ X n=1 |gn||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{1/3, rf}, where rf is given as in Theorem 2.9. Similar results on Bohr phenomenon for the class S∗ c (φ) hold for the class S∗

Corollary 2.14. The class S∗ c ((1 + sz)2) and S∗ cf ((1 + sz)2)
satisfies the Bohr inequality (2.10) for |z| = r ≤rf, where 0 < rf < 1/3 and rf is the root of the equation (2.15) r exp  s  2r + sr2

Corollary 2.16. For φ(z) = α + (1 −α)ez with 0 ≤α < 0.05284, the class S∗ c (φ) satisfies the Bohr phenomenon (2.10) for |z| = r ≤rf, where 0 < rf < 1/3 . The rdius rf is the best possible.

Corollary 2.17. Let φ(z) = ((1 + z)/(1 −z))α with 0 < α ≤1. Also assume h(1/3) > −h(−1), where h(r) = r exp   r Z 0 1+t 1−t
α −1 t dt 

Corollary 2.18. Let φ(z) = (1 + (1 −2γ)z) /(1 −z) with 0 ≤γ < 1/2. Then each f ∈S∗ c ((1 + (1 −2γ)z) /(1 −z)) satisfies the inequality (2.10) for |z| = r ≤rf, where 0 < rf < 1/3 and rf is the root of (2.19) r + 2r1/(2(1−γ)) −1 = 0. The radius rf is the best possible.

Corollary 2.20. If φ(z) = (1 + Az)/(1 + Bz) with −1 ≤B < A ≤1, then (i) when B = 0, every function f ∈S∗ c ((1 + Az)/(1 + Bz)) satisfies the inequal- ity (2.10) for |z| = r ≤rf, where 0 < rf < 1/3 and rf is the unique root of (2.21) reAr = e−A, provided A ≥(3/4)ln 3. The radius rf is the best possible. (ii) When B ̸= 0, every function f ∈S∗ c ((1 + Az)/(1 + Bz)) satisfies the in- equality (2.10) for |z| = r ≤rf, where 0 < rf < 1/3 and rf is the unique root of (2.22) r (1 + Br) A−B

Theorem 2.23. Let f ∈Cc(φ) be of the form (1.5). Then (2.24) |z| + ∞ X n=2 |an||z|n ≤d(f(0), ∂f(D))

Theorem 2.25. Let f ∈Cs(φ) be of the form (1.5). Then (2.26) |z| + ∞ X n=2 |an||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{1/3, rf} and rf is the smallest positive root of Rs(r) = Ls(1) in (0, 1), where Rs(r) := r Z 0 1 s

Lemma 1.19, we obtain Mq(r) ≤Mf(r) for |z| = r ≤1/3. Hence from (2.3), we get P∞ n=1 |qn||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{1/3, rf}. □

Lemma 1.19, we obtain Mg(r) ≤Mf(r) for |z| = r ≤1/3. Hence from (2.10), we obtain P∞ n=1 |gn||z|n ≤d(f(0), ∂f(D)) for |z| = r ≤min{1/3, rf}. □