Some Bohr-Type Inequalities for Bounded Analytic Functions

the value 1/3 is the best possible, and the inequality (1) is called classical Bohr inequality. In 1914, Bohr originally established the inequality (1) for |z|< 1/6 [1]. But subsequently later, Wiener, Riesz, and Schur independently proved the inequality (1) for |z|< 1/3, and the value is proved to be sharp [2–4]. Bohr phenomenon and Bohr radius problem are very interesting. Many generalized forms are studied and a lot of Bohr-type inequalities are obtained. Kayumov considered the Bohr-type inequalities for odd analytic functions and some other classes of analytic function in [5]. Bhowmik and Das [6] studied Bohr’s phenomenon for subordinating families of certain univalent functions. Huang et al. [7] refined the Bohr inequality by allowing Schwarz function in place of the initial variable of the functions. Hu et al. [8, 9] established some Bohr inequalities with one parameter or involving convex combination. Abu-Muhanna [10] investigated the Bohr phenomenon for the class of analytic functions from the unit disk into the punctured unit disk. Some authors considered the Bohr phenomenon for functions defined on other domains, such as concave-wedge domain [11, 12], convex domain [13], and the exterior of a compact domain [14]. *e Bohr-type inequalities were also considered in other branches of mathematics. *e analogous Bohr’s radius was also studied for K-quasiconformal harmonic mappings by Liu and Ponnusamy [15]. Djakov and Ramanujan [16], Aizenberg [17], and Bayart et al. [18] generalized the Bohr inequality to the case of higher dimensions. Abu-Muhanna and Gunatillake [19] considered Bohr phenomenon in weighted Hardy-Hilbert spaces. Blasco [20] obtained the Bohr’s radius of a Banach space. In [21], Paulsen explored Bohr’s radius problem in Banach algebras. For more discussion on the Bohr-type inequalities, we can refer to [22–25]. We recall some results as follows.


Introduction and Preliminaries
Let H denote the class of analytic functions f(z) � ∞ k�0 a k z k defined in the unit disk D ≔ z ∈ C: |z| < 1 { } such that |f(z)| < 1 in D. e Bohr's theorem states that if f(z) � Σ ∞ k�0 a k z k ∈ H, then ∞ k�0 a k ‖z k ≤ 1, for |z| ≤ the value 1/3 is the best possible, and the inequality (1) is called classical Bohr inequality. In 1914, Bohr originally established the inequality (1) for |z| < 1/6 [1]. But subsequently later, Wiener, Riesz, and Schur independently proved the inequality (1) for |z| < 1/3, and the value is proved to be sharp [2][3][4]. Bohr phenomenon and Bohr radius problem are very interesting. Many generalized forms are studied and a lot of Bohr-type inequalities are obtained. Kayumov considered the Bohr-type inequalities for odd analytic functions and some other classes of analytic function in [5]. Bhowmik and Das [6] studied Bohr's phenomenon for subordinating families of certain univalent functions. Huang et al. [7] refined the Bohr inequality by allowing Schwarz function in place of the initial variable of the functions. Hu et al. [8,9] established some Bohr inequalities with one parameter or involving convex combination. Abu-Muhanna [10] investigated the Bohr phenomenon for the class of analytic functions from the unit disk into the punctured unit disk. Some authors considered the Bohr phenomenon for functions defined on other domains, such as concave-wedge domain [11,12], convex domain [13], and the exterior of a compact domain [14]. e Bohr-type inequalities were also considered in other branches of mathematics. e analogous Bohr's radius was also studied for K-quasiconformal harmonic mappings by Liu and Ponnusamy [15]. Djakov and Ramanujan [16], Aizenberg [17], and Bayart et al. [18] generalized the Bohr inequality to the case of higher dimensions. Abu-Muhanna and Gunatillake [19] considered Bohr phenomenon in weighted Hardy-Hilbert spaces. Blasco [20] obtained the Bohr's radius of a Banach space. In [21], Paulsen explored Bohr's radius problem in Banach algebras. For more discussion on the Bohr-type inequalities, we can refer to [22][23][24][25].
We recall some results as follows.
where α is the positive root of the equation e radius α is the best possible. Moreover, where β is the positive root of the equation e radius β is the best possible.
Theorem 2 (see [27]). Suppose f(z) � ∞ k�0 a k z k ∈ H, and s ∈ N, then where c is the positive root of the equation e radius c is the best possible.
In order to establish our main results, we need the following lemmas, which will play the key role in proving the main results of this paper.
Lemma 1 (see [27]) (Schwarz-Pick lemma). Suppose and the equality holds for distinct z 1 , z 2 ∈ D if and only if f(z) is a Möbius transformation.
In particular, and the equality holds for some z ∈ D, if and only if f(z) is a Möbius transformation.
e proof is simple. We omit it.

Main Results
Theorem 5. Suppose f(z) � ∞ k�0 a k z k ∈ H, a ≔ |a 0 | and ω m ∈ S m , ψ n ∈ S n for m, n ∈ N. en, for λ ∈ (0, +∞) and N, s ∈ N, we have for |z| � r ≤ R λ,N,m,n,s , where R λ,N,m,n,s is the unique root in (0, 1) of the equation and the radius R λ,N,m,n,s is the best possible.
Proof. By the Schwarz lemma and the Schwarz-Pick lemma, respectively, we obtain en, by Lemma 2, we obtain 2
In eorem 5, setting s � 1, then we have Corollary 1. □ for |z| � r ≤ R λ,,N,m,n , where R λ,N,m,n is the unique root in (0, 1) of the equation and the radius R λ,N,m,n is the best possible.
Theorem 6. Suppose f(z) � ∞ k�0 a k z k ∈ H, a ≔ |a 0 | and ω m ∈ S m , ψ n ∈ S n for m, n ∈ N. en, for λ ∈ (0, +∞) and N ∈ N, we have for |z| � r ≤ R * λ,N,m,n,s , where R * λ,N,m,n,s is the unique root in (0, 1) of the equation and the radius R * λ,N,m,n,s is the best possible.
□ Corollary 2. Suppose f(z) � ∞ k�0 a k z k ∈ H, a ≔ |a 0 | and ω m ∈ S m , ψ n ∈ S n for m, n ∈ N. en, for λ ∈ (0, +∞) and N ∈ N, we have for |z| � r ≤ R * λ,N,m,n , where R λ,N,m,n is the unique root in (0, 1) of the equation and the radius R * λ,N,m,n is the best possible.
Theorem 7. Suppose f(z) � ∞ k�0 a k z k ∈ H, a ≔ |a 0 |, and ω m ∈ S m for m ∈ N. en, for λ ∈ (0, +∞) and N ∈ N, we have ) of the equation and the radius R λ,N,m is the best possible.
Theorem 8. Suppose f(z) � ∞ k�0 a k z k ∈ H, a ≔ |a 0 |, and ω m ∈ S m , for m ∈ N. en, for t ∈ [0, 1), we have and the radius R t,m is the best possible.
Proof. Inequalities (43) and Lemmas 1, 2 and 4 lead to that We just need to show that D m (a, r, t) ≤ 1 holds for r ≤ R t,m . at is to prove D(a, r, t) ≤ 0 for r ≤ R t,m , where en, we divide it into two cases to discuss.
Now, we show that the radius R t,m is the best possible. Let a ∈ [0, 1); we still consider the functions ω m (z), f(z) as in (48). Taking z � r ∈ (0, 1), then the left side of inequality (53) reduces to 1 − a 2 2 a 2k− 2 r 2mk ⎡ ⎣ ⎤ ⎦ � t a + r m 1 + ar m +(1 − t) a + 1 − a 2 r m 1 − r m ≔ D * m (a, r, t). (58) Now, we just need to show that if r > R t,m , then there exists an a ∈ [0, 1) such that D * m (a, r, t) is greater than 1. at is to prove l(a, r, t) > 0 for r > R t,m , where  (60) Next, we divide it into two cases to discuss.

Conclusion
We establish some new versions of Bohr-type inequalities with one parameter or involving convex combination for bounded analytic functions of Schwarz function, and we conclude that all of the corresponding Bohr radii in the paper are exact. ese inequalities generalize some earlier results on the Bohr radius problem.
Data Availability e author declares that this research is purely theoretical and does not associate with any data.

Conflicts of Interest
e author declares that there are no conflicts of interest.