Bohr inequalities for certain integral operators
aa r X i v : . [ m a t h . C V ] A ug BOHR INEQUALITIES FOR CERTAIN INTEGRAL OPERATORS
SHANKEY KUMAR AND SWADESH KUMAR SAHOO
Abstract.
In this article, we determine sharp Bohr-type radii for certain complex integraloperators defined on a set of bounded analytic functions in the unit disk. Introduction
We denote by D := { z ∈ C : | z | < } the unit disk in the complex plane. Let H be theclass of all analytic functions defined on D . Setting B = { f ∈ H : | f ( z ) | ≤ } . Let us firsthighlight a remarkable result of Bohr [13] that opens up a new type of research problems ingeometric function theory, which states that “ If f ( z ) = P ∞ n =0 a n z n ∈ B , then ∞ X n =0 | a n | r n ≤ , for r ≤ / and the constant / cannot be improved. ” The quantity 1 / Bohrradius for the class B . Moreover, for functions in B := { f ∈ B | f (0) = 0 } , Bombieri [14]found the Bohr radius, which is 1 / √ B can be foundin [35].In [7], Ali et al. brought into the notice of the Bohr radius problem for the odd ana-lytic functions, which is settled by Kayumov and Ponnusamy in [19]. Also, Kayumov andPonnusamy [21] generalized the problem of the Bohr radius for the odd analytic functions.Bhowmik and Das [10] studied the Bohr radius for families of certain analytic univalent(one-to-one) functions. In [9], the Bohr phenomenon is discussed for the functions in Hardyspaces. The study of the Bohr radius of the Bloch functions discussed in [23]. In [8, 34], au-thors studied the Bohr phenomenon for a quasi-subordination family of functions. Recently,Bhowmik and Das [12] studied the Bohr radius for derivatives of analytic functions. To findmore achievements in this context, one may see the papers [1–4, 11, 20, 22, 26–29] and thereferences there in. Also, the survey article [5] and the references cited in it are useful in thisdirection.A natural question arises “can we find Bohr radius for certain complex integral operatorsdefined either on the class B or B ?” . This idea has been initiated first for the classical Ces´arooperator in [18]. As our results of this paper are motivated by [18], here first we recall thedefinition of the Ces´aro operator followed by statement of the result on absolute sum of theseries representation of the operator. The Ces´aro operator is studied in [16] (see, for more Mathematics Subject Classification.
Primary: 30H05, 35A22; Secondary: 30A10, 30C80.
Key words and phrases.
Bohr inequality, Integral transforms. information, [37] and [38]) which is defined as(1.1) T [ f ]( z ) := Z f ( tz )1 − tz dt = ∞ X n =0 n + 1 n X k =0 a k ! z n , where f ( z ) = P ∞ n =0 a n z n is analytic in D . Also, a generalized form of the Ces´aro operator isstudied in [6].As noted in [18], | T [ f ]( z ) | ≤ r log 11 − r for each | z | = r <
1. On the other hand, from (1.1), we also have the obvious estimate | T [ f ]( z ) | ≤ ∞ X n =0 (cid:18) n + 1 n X k =0 | a k | (cid:19) | z | n , the absolute sum of the series (1.1). However, if | z | = r <
1, Kayumov et al. [18] obtained thesharp radius r for which this absolute sum has the same upper bound (1 /r ) log(1 / (1 − r )). Thiswas important to study, as in general, a convergent series need not be absolutely convergent.Indeed, they established Theorem A. If f ( z ) = P ∞ n =0 a n z n ∈ B , then ∞ X n =0 (cid:18) n + 1 n X k =0 | a k | (cid:19) r n ≤ r log 11 − r for r ≤ R = 0 . . . . . Here the number R is the positive root of the equation x − − x ) log 11 − x = 0 that cannot be improved. Motivated by Theorem A, in this paper, we study the Bohr radius problem for the β -Ces´arooperator ( β >
0) defined by T β [ f ]( z ) := ∞ X n =0 (cid:18) n + 1 n X k =0 Γ( n − k + β )Γ( n − k + 1)Γ( β ) a k (cid:19) z n = Z f ( tz )(1 − tz ) β dt, z ∈ D , and for the Bernardi operator defined as L γ [ g ]( z ) := ∞ X n = m a n n + γ z n = Z g ( zt ) t γ − dt, for g ( z ) = P ∞ n = m a n z n and γ > − m , here m ≥ OHR INEQUALITIES FOR CERTAIN INTEGRAL OPERATORS 3 Main results
Note that the β -Ces´aro operator T β ( β >
0) is a natural generalization of the Ces´arooperator T defined by (1.1) and indeed, we have T = T . For f ∈ B and β >
0, anelementary estimation of the integral in absolute value gives us the sharp inequality | T β [ g ]( z ) | ≤ r (cid:20) − (1 − r ) − β − β (cid:21) , if β = 1 , r log 11 − r, if β = 1 , for each | z | = r <
1. In this line, similar to Theorem A, our first main result is the following.
Theorem 2.1.
For f ( z ) = P ∞ n =0 a n z n ∈ B and < β = 1 , we have ∞ X n =0 (cid:18) n + 1 n X k =0 Γ( n − k + β )Γ( n − k + 1)Γ( β ) | a k | (cid:19) r n ≤ r (cid:20) − (1 − r ) − β − β (cid:21) , for r ≤ R ( β ) , where R ( β ) is the positive root of the equation − (1 − x ) − β ]1 − β − − x ) − β − β = 0 . The radius R ( β ) cannot be improved. Here, it is easy to observe that if we take the limit β → Remark 2.1.
Another form of the β -Ces´aro operator of a normalized analytic function g ( z ) = P ∞ n =1 b n z n in D has been studied in the literature (see [25]): C β [ g ]( z ) = Z g ( tz ) t (1 − tz ) β dt = ∞ X n =0 (cid:18) n + 1 n X k =0 Γ( n − k + β )Γ( n − k + 1)Γ( β ) b k +1 (cid:19) z n +1 , z ∈ D , for β >
0. This version of the β -Ces´aro operator was initially considered to study its bound-edness, compactness, and spectral properties, and more recently its univalency propertieswere investigated in [24]. To study its Bohr radius problem, it is necessary for us to assumethat g ( z ) = P ∞ n =1 b n z n ∈ B . An easy calculation gives us the sharp inequality, for g ∈ B and β > | C β [ g ]( z ) | ≤ − (1 − r ) − β − β , if β = 1 , log 11 − r, if β = 1 , for each | z | = r <
1. It is well-known by the Schwarz lemma that if g ( z ) = P ∞ n =1 b n z n ∈ B then we can write g ( z ) = zh ( z ) for h ( z ) = P ∞ n =0 b n +1 z n ∈ B . So, we have C β [ g ]( z ) = ∞ X n =0 (cid:18) n + 1 n X k =0 Γ( n − k + β )Γ( n − k + 1)Γ( β ) b k +1 (cid:19) z n +1 = zT β [ h ]( z ) . SHANKEY KUMAR AND S. K. SAHOO
Now, by using Theorem 2.1 we obtain ∞ X n =0 (cid:18) n + 1 n X k =0 Γ( n − k + β )Γ( n − k + 1)Γ( β ) | b k +1 | (cid:19) r n +1 ≤ − (1 − r ) − β − β , < β = 1 , for r ≤ R ( β ). Here R ( β ) is the positive root of the equation3[1 − (1 − x ) − β ]1 − β − − x ) − β − β = 0that cannot be improved. Recall that the operator C has been considered in [17, 24, 25, 33]for various aspects. Moreover, in the limit β →
1, we can indeed obtain the Bohr radiusproblem: If g ( z ) = P ∞ n =1 b n z n ∈ B then ∞ X n =0 (cid:18) n + 1 n X k =0 | b k +1 | (cid:19) r n +1 ≤ log 11 − r for r ≤ R = 0 . . . . . The number R is the positive root of the equation x − − x ) log 11 − x = 0 that cannot be improved. This remark observes that the Bohr radii for the operators T β and C β are almost same, but up to an extra factor 1 /r . (cid:3) Similar to the Bohr-type radius problem of the operator T β , β >
0, we also study the Bohrradius of the absolute series of the Bernardi operator [30, P. 11] (see also [32]) defined by L γ [ f ]( z ) := ∞ X n = m a n n + γ z n = Z f ( zt ) t γ − dt, for f ( z ) = P ∞ n = m a n z n and γ > − m , here m ≥ L γ [ f ] is analyticin D and several properties of L γ [ f ] when m = 1 (with a normalization) are well-known (see,for instance [30–32]).It is easy to calculate the following sharp bound | L γ [ f ]( z ) | ≤ m + γ r m , | z | = r < f ( z ) = P ∞ n = m a n z n . Corresponding to the above inequality, we obtain the following result. Theorem 2.2.
Let γ > − m . If f ( z ) = P ∞ n = m a n z n ∈ B , then ∞ X n = m | a n | n + γ r n ≤ m + γ r m for r ≤ R ( γ ) . Here, R ( γ ) is the positive root of the equation x m m + γ − ∞ X n = m +1 x n n + γ = 0 that cannot be improved. OHR INEQUALITIES FOR CERTAIN INTEGRAL OPERATORS 5
Letting γ = 1 in the Bernardi operator L γ , we obtain the well-known Libera operator [30,32]defined as L [ f ]( z ) := Z f ( zt ) dt = ∞ X n =0 a n n + 1 z n . The multiplication of z in the Libera operator L gives the integral I [ f ]( z ) := ∞ X n =0 a n n + 1 z n +1 = Z z f ( w ) dw, | z | < . It is easy to check that | L [ f ]( z ) | ≤ | I [ f ]( z ) | ≤ r, | z | = r. As a special case of Theorem 2.1 ( γ = 1 and m = 0), we get the Bohr radius for the Liberaoperator as well as for the operator I as follows. Corollary 2.3. If f ( z ) = P ∞ n =0 a n z n ∈ B , then ∞ X n =0 | a n | n + 1 r n ≤ , for r ≤ R with R = 0 . . . . , the positive root of the equation x + 2 log(1 − x ) = 0 . Here, R is the best possible. Also, the Alexander operator [15, 24, 25, 30] J [ g ]( z ) := Z g ( zt ) t dt = ∞ X n =1 b n n z n , for g ( z ) = P ∞ n =1 b n z n , extensively studied in the univalent function theory. We have sharpbound | J [ g ]( z ) | ≤ r for each | z | = r <
1, since | g ( zt ) /t | ≤ g ∈ B we can obtain an element h ∈ B such that g ( z ) = zh ( z ). So, wehave the following result as a consequence of Corollary 2.3. Corollary 2.4. If g ( z ) = P ∞ n =1 b n z n ∈ B , then ∞ X n =1 | b n | n r n ≤ r, for r ≤ R . Here, R = 0 . · · · is the positive root of the equation x + 2 log(1 − x ) = 0 thatcannot be improved. In the next section, we discuss the proofs of Theorems 2.1 and 2.2.
SHANKEY KUMAR AND S. K. SAHOO Proofs of the main results
Proof of Theorem 2.1.
First we define(3.1) T fβ ( r ) := ∞ X n =0 (cid:18) n + 1 n X k =0 Γ( n − k + β )Γ( n − k + 1)Γ( β ) | a k | (cid:19) r n , where f ( z ) = P ∞ n =0 a n z n ∈ B , 0 < β = 1 and r = | z | <
1. Setting | a | := a and let a <
1. ByWiener’s estimate we know that | a n | ≤ − a for n ≥
1. This yields T fβ ( r ) ≤ a ∞ X n =0 (cid:18) n + 1 Γ( n + β )Γ( n + 1)Γ( β ) (cid:19) r n + (1 − a ) ∞ X n =1 (cid:18) n + 1 n X k =1 Γ( n − k + β )Γ( n − k + 1)Γ( β ) (cid:19) r n . The above inequality is equivalent to T fβ ( r ) ≤ ar Z r − t ) β dt + (1 − a ) r Z r t (1 − t ) β +1 dt = ( a + a − r Z r − t ) β dt + (1 − a ) r Z r − t ) β +1 dt. It follows that T fβ ( r ) ≤ r (cid:20) ( a + a − − (1 − r ) − β ]1 − β + (1 − a )[(1 − r ) − β − β (cid:21) := φ ( a ) . Differentiation of the function φ with respect to a gives us φ ′ ( a ) = 1 r (cid:20) (2 a + 1)[1 − (1 − r ) − β ]1 − β − a [(1 − r ) − β − β (cid:21) and so φ ′′ ( a ) = 1 r (cid:20) − (1 − r ) − β ]1 − β − − r ) − β − β (cid:21) . It is easy to see that φ ′′ ( a ) ≤ a ∈ [0 ,
1) and r ∈ [0 , φ ′ ( a ) ≥ φ ′ (1). Here φ ′ (1) = 1 r (cid:20) − (1 − r ) − β ]1 − β − − r ) − β − β (cid:21) ≥ r ≤ R ( β ), where R ( β ) is the positive root of the equation3[1 − (1 − x ) − β ]1 − β − − x ) − β − β = 0 . Then φ ( a ) is an increasing function of a , for r ≤ R ( β ). It implies that φ ( a ) ≤ φ (1) = 1 r (cid:20) − (1 − r ) − β − β (cid:21) , for r ≤ R ( β ). It is easy to observe that R ( β ) <
1. This completes the first part of thetheorem.To conclude the final part, we consider the function φ a ( z ) = z − a − az = − a + (1 − a ) ∞ X n =1 a n − z n , OHR INEQUALITIES FOR CERTAIN INTEGRAL OPERATORS 7 where z ∈ D and a ∈ [0 , T φ a β ( r ) = ar (cid:20) − (1 − r ) − β − β (cid:21) + (1 − a ) ∞ X n =1 (cid:18) a n − n + 1 n X k =1 Γ( n − k + β )Γ( n − k + 1)Γ( β ) (cid:19) r n = ar (cid:20) − (1 − r ) − β − β (cid:21) + (1 − a ) r Z r t (1 − at )(1 − t ) β dt. We can rewrite the last expression as(3.2) T φ a β ( r ) = 1 r (cid:20) − (1 − r ) − β − β (cid:21) − (1 − a ) r " − (1 − r ) − β ]1 − β − − r ) − β − β + N a ( r ) , where N a ( r ) = 2(1 − a ) r " [1 − (1 − r ) − β ]1 − β − [(1 − r ) − β − β + (1 − a ) r Z r t (1 − at )(1 − t ) β dt. Expressing N a ( r ) into its summation form, we have N a ( r ) = ∞ X n =0 n + 1 − (1 − a ) a Γ( n + β )Γ( n + 1)Γ( β ) − − a ) Γ( n + β + 1)Γ( n + 1)Γ( β + 1)+ (1 − a ) a n X m =0 Γ( n − m + β )Γ( n − m + 1)Γ( β ) a m ! r n . By using the identity n X m =0 Γ( n − m + β )Γ( n − m + 1)Γ( β ) = Γ( n + β + 1)Γ( n + 1)Γ( β + 1) , we can get that N a ( r ) = O ((1 − a ) ), as a tends to 1. Further, a simple computation showsthat for r > R ( β ) the quantity3[1 − (1 − r ) − β ]1 − β − − r ) − β − β < . After using these observations in (3.2) we conclude that R ( β ) cannot be improved. Thiscompletes the proof. (cid:3) Proof of Theorem 2.2.
Given that f ( z ) = P ∞ n = m a n z n ∈ B . We set the notation(3.3) L f ( r ) := ∞ X n = m | a n | n + γ r n . The Schwarz lemma gives f ( z ) = z m h ( z ), where h ( z ) = P ∞ n = m a n z n − m . Denoting by a := | a m | < | a n | ≤ (1 − a ) for n ≥ m + 1 in (3.3), we obtain thefollowing inequality L f ( r ) ≤ am + γ r m + (1 − a ) ∞ X n = m +1 n + γ r n := ψ ( a ) . SHANKEY KUMAR AND S. K. SAHOO
It is easy to see that ψ ′′ ( a ) = − ∞ X n = m +1 n + γ r n ≤ . Thus, ψ ′ ( a ) ≥ ψ ′ (1) = 1 m + γ r m − ∞ X n = m +1 n + γ r n ≥ , for r ≤ R ( γ ), where R ( γ ) is the positive root of the equation1 m + γ r m − ∞ X n = m +1 n + γ r n = 0 . Hence, ψ ( a ) is an increasing function of a for r ≤ R ( γ ). This gives that ∞ X n = m | a n | n + γ r n ≤ m + γ r m , for r ≤ R ( γ ) . Also, a simple observation gives R ( γ ) < R ( γ ) is the best possible bound, we consider the function ψ a ( z ) = z m z − a − az = − az m + (1 − a ) ∞ X n =1 a n − z n + m , where z ∈ D and a ∈ [0 , L ψ a ( r ) = am + γ r m + (1 − a ) ∞ X n = m +1 a n − n + γ r n with the help of (3.3), which is equivalent to(3.4) L ψ a ( r ) = 1 m + γ r m − (1 − a ) m + γ r m − ∞ X n = m +1 n + γ r n ! + M a ( r ) , where M a ( r ) = 2( a − ∞ X n = m +1 n + γ r n + (1 − a ) ∞ X n = m +1 a n − n + γ r n . Letting a →
1, we obtain M a ( r ) = ∞ X n = m +1 a −
1) + (1 − a ) a n − n + γ r n = O ((1 − a ) ) . Further, the quantity 1 m + γ r m − ∞ X n = m +1 n + γ r n < r > R ( γ ). These facts in (3.4) gives that R ( γ ) cannot be improved and the proofis complete. Acknowledgement.
The authors thank Professor S. Ponnusamy for bringing some of theBohr radius papers including [18] to their attention and useful discussion on this topic. The
OHR INEQUALITIES FOR CERTAIN INTEGRAL OPERATORS 9 work of the first author is supported by CSIR, New Delhi (Grant No: 09/1022(0034)/2017-EMR-I).
Conflict of Interests.
The authors declare that there is no conflict of interests regardingthe publication of this paper.
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