# On Certain Classes of Harmonicp-Valent Functions Defined by an Integral Operator

Authors: T. M. Seoudy
Journal: International Journal of Analysis
 (2013)

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## Abstract

We obtain coefficient characterization, extreme points, and distortion bounds of certain classes of harmonicp-valent functions defined by an integral operator.


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## Body

## 1. Introduction

A continuous complex-valued functionf=u+iv defined in a simply connected complex domain D is said to be harmonic in D if both u and v are real harmonic in D. In any simply connected domain, we can  write
(1)f=h+g¯,
where h and g are analytic in D. We call h the analytic part and g the coanalytic part of f. A necessary and sufficient condition for f to be locally univalent and sense preserving in D is that |h′(z)|>|g′(z)| in D (see [1]).Denote bySH the class of functions f of the form (1) that are harmonic univalent and sense preserving in the unit disc U={z:|z|<1} for which f(0)=fz(0)-1=0.Recently, Jahangiri and Ahuja [2] defined the class ℋp(p∈ℕ={1,2,3,…}), consisting of all p-valent harmonic functions f=h+g¯ that are sense preserving in U and h, and g are of the form
(2)h(z)=zp+∑k=p+1∞‍akzk,g(z)=∑k=p∞‍bkzk,|bp|<1.Forf=h+g¯ given by (2), we define the modified p-valent Salagean integral operator Ip,λn of f (see [3] and also [4] when p=1) as follows:
(3)Ip,λnf(z)=Ip,λnh(z)+(-1)nIp,λng(z)¯,
where
(4)Ip,λnh(z)=zp+∑k=p+1∞‍(pp+λ(k-p))nakzk12(p∈ℕ;λ>0;n∈ℕ0=ℕ∪{0}),Ip,λng(z)=∑k=p∞‍(pp+λ(k-p))nbkzk1&2(p∈ℕ;λ>0;n∈ℕ0).Forp∈ℕ,λ>0,n∈ℕ0,0≤α<1, and z∈U, we let ℋp,λ(n;α) denote the family of harmonic functions f of the form (2) such that
(5)Re{Ip,λnf(z)Ip,λn+1f(z)}>α,
where Ip,λnf is defined by (3).We let the subclassℋp,λ-(n;α) consists of harmonic functions fn=h+g¯n in ℋp,λ(n;α) so that h and gn are of the form
(6)h(z)=zp-∑k=p+1∞‍akzk,gn(z)=(-1)n∑k=p∞‍bkzk,12345678945612348745ak,bk≥0.We note thatℋp,1-(n;α)=ℋp-(n;α), where the class ℋp-(n;α) was defined and studied by Cotirla [5].In this paper, we obtain coefficient characterization of the classesℋp,λ(n;α) and ℋp,λ-(n;α). We also obtain extreme points and distortion bounds for functions in the class ℋp,λ-(n;α).


## 2. Coefficient Characterization

Unless otherwise mentioned, we assume throughout this paper thatp∈ℕ,n∈ℕ0,0≤α<1,ap=1, and λ>0. We begin with a sufficient condition for functions in ℋp,λ(n;α).Theorem 1.
Letf=h+g¯ so that h and g are given by (2). Furthermore, let
(7)∑k=p∞‍{Ψp,λ(n,k,α)|ak|+Φp,λ(n,k,α)|bk|}≤2,
where
(8)Ψp,λ(n,k,α)=((pp+λ(k-p))n-α(pp+λ(k-p))n+1)×(1-α)-1,(9)Φp,λ(n,k,α)=((pp+λ(k-p))n+α(pp+λ(k-p))n+1)×(1-α)-1.
Then, f is sense preserving in U and f∈ℋp,λ(n;α).Proof.
According to (2) and (3), we only need to show that
(10)Re{Ip,λnf(z)-αIp,λn+1f(z)Ip,λn+1f(z)}≥0(z∈U).
It follows that
(11)Re{Ip,λnf(z)-αIp,λn+1f(z)Ip,λn+1f(z)}=Re{((1-α)zp+∑k=p+1∞×[‍(pp+λ(k-p))n-α(pp+λ(k-p))n+1]akzk)×(zp+∑k=p+1∞‍(pp+λ(k-p))n+1akzk+(-1)n+1∑k=p∞‍(pp+λ(k-p))n+1b-kz-k)-1+((-1)n∑k=p∞[(pp+λ(k-p))n‍+α(pp+λ(k-p))n+1]b-kz-k)×(zp+∑k=p+1∞‍(pp+λ(k-p))n+1akzk+(-1)n+1∑k=p∞‍(pp+λ(k-p))n+1b-kz-k)-1}=Re{((1-α)+∑k=p+1∞‍×[(pp+λ(k-p))n-α(pp+λ(k-p))n+1]akzk-p)×(1+∑k=p+1∞‍(pp+λ(k-p))n+1akzk-p+(-1)n+1∑k=p∞‍(pp+λ(k-p))n+1b-kz-kz-p)-1+((-1)n∑k=p∞[(pp+λ(k-p))n‍+α(pp+λ(k-p))n+1]×b-kz-kz-p)×(1+∑k=p+1∞‍(pp+λ(k-p))n+1×akzk-p+(-1)n+1∑k=p∞‍(pp+λ(k-p))n+1×b-kz-kz-p)-1}=Re{1-α+A(z)1+B(z)}.
For z=reiθ, we have
(12)A(reiθ)=∑k=p+1∞‍[(pp+λ(k-p))n-α(pp+λ(k-p))n+1]akrk-pei(k-p)θ+(-1)n∑k=p∞‍[(pp+λ(k-p))n+α(pp+λ(k-p))n+1]×b-krk-pe-i(k+p)θ,B(reiθ)=∑k=p+1∞‍(pp+λ(k-p))n+1akrk-pei(k-p)θ+(-1)n+1∑k=p∞‍(pp+λ(k-p))n+1b-krk-pe-i(k+p)θ.
Setting that
(13)1-α+A(z)1+B(z)=(1-α)1+w(z)1-w(z),
the proof will be complete if we can show that |w(z)|<1. Using the condition (7), we can write
(14)|w(z)|=|A(z)-(1-α)B(z)A(z)+(1-α)B(z)+2(1-α)|=|(∑k=p+1∞‍[(pp+λ(k-p))n-(pp+λ(k-p))n+1]akrk-pei(k-p)θ)×(2(1-α)+∑k=p+1∞‍ckakrk-pei(k-p)θ+(-1)n∑k=p∞‍dkb-krk-pe-i(k+p)θ)-1+((-1)n∑k=p∞‍[(pp+λ(k-p))n+α(pp+λ(k-p))n+1]×b-krk-pe-i(k+p)θ)×(2(1-α)+∑k=p+1∞‍ckakrk-pei(k-p)θ+(-1)n∑k=p∞‍dkb-krk-pe-i(k+p)θ)-1|≤|(∑k=p+1∞‍[(pp+λ(k-p))n-(pp+λ(k-p))n+1]|ak|rk-p)×(2(1-α)-∑k=p+1∞‍ck|ak|rk-p-∑k=p∞‍dk|bk|rk-p)-1+(∑k=p∞‍[(pp+λ(k-p))n+(pp+λ(k-p))n+1]|bk|rk-p)×(2(1-α)-∑k=p+1∞‍ck|ak|rk-p-∑k=p∞‍dk|bk|rk-p)-1|=|(∑k=p+1∞‍[(pp+λ(k-p))n-(pp+λ(k-p))n+1]|ak|rk-p)×(4(1-α)-∑k=p∞‍{ck|ak|+dk|bk|}rk-p)-1+(∑k=p∞[(pp+λ(k-p))n‍+(pp+λ(k-p))n+1]|bk|rk-p)×(4(1-α)-∑k=p∞‍{ck|ak|+dk|bk|}rk-p)-1|<|(∑k=p+1∞‍[(pp+λ(k-p))n-(pp+λ(k-p))n+1]|ak|)×(4(1-α)-∑k=p∞‍{ck|ak|+dk|bk|})-1+(∑k=p∞‍[(pp+λ(k-p))n+(pp+λ(k-p))n+1]|bk|)×(4(1-α)-∑k=p∞‍{ck|ak|+dk|bk|})-1|≤1,
where
(15)ck=(pp+λ(k-p))n+(1-2α)(pp+λ(k-p))n+1,dk=(pp+λ(k-p))n-(1-2α)(pp+λ(k-p))n+1.
The harmonic functions are as follows:(16)f(z)=zp+∑k=p+1∞‍1Ψp,λ(n,k,α)xkzk123+∑k=p∞‍1Φp,λ(n,k,α)ykzk¯,
where ∑k=p+1∞|xk|+∑k=p∞|yk|=1 show that the coefficient bound given by (7) is sharp. The functions of the form (8) are in the class ℋp,λ(n;α) because
(17)∑k=p∞‍{Ψp,λ(n,k,α)|ak|+Φp,λ(n,k,α)|bk|}=1+∑k=p+1∞‍|xk|+∑k=p∞‍|yk|=2.
This completes the proof of Theorem1.In the following theorem, it is shown that the condition (7) is also necessary for functions fn=h+g¯n, where h and gn are of the form (6).Theorem 2.
Letfn=h+g¯n, where h and gn are given by (6). Then, fn∈ℋp,λ-(n;α) if and only if
(18)∑k=p∞‍{Ψp,λ(n,k,α)ak+Φp,λ(n,k,α)bk}≤2,
where Ψp,λ(n,k,α) and Φp,λ(n,k,α) are given by (8) and (9), respectively.Proof.
Sinceℋp,λ-(n;α)⊂ℋp,λ(n;α), we only need to prove the “only if” part of the theorem. To this end, for functions fn=h+g¯n, where h and gn are given by (6), we notice that the condition Re{Ip,λnf(z)/Ip,λn+1f(z)}>αis equivalent to(19)Re{((1-α)zp-∑k=p+1∞‍×[(pp+λ(k-p))n-α(pp+λ(k-p))n+1]akzk)×(zp-∑k=p+1∞‍(pp+λ(k-p))n+1akzk+(-1)2n∑k=p∞‍(pp+λ(k-p))n+1bkz-k)-1+((-1)2n-1∑k=p∞‍×[(pp+λ(k-p))n+α(pp+λ(k-p))n+1]bkz-k)×(zp-∑k=p+1∞‍(pp+λ(k-p))n+1akzk+(-1)2n∑k=p∞‍(pp+λ(k-p))n+1b-kz-k)-1}≥0.
The previous required condition (19) must hold for all values of z in U. Upon choosing the values of z on the positive real axis where 0≤z=r<1, we must have
(20)((1-α)-∑k=p+1∞‍[(pp+λ(k-p))n-α(pp+λ(k-p))n+1]akrk-p)×(1-∑k=p+1∞‍(pp+λ(k-p))n+1akrk-p+∑k=p∞‍(pp+λ(k-p))n+1bkrk-p)-1+(-∑k=p∞‍[(pp+λ(k-p))n+α(pp+λ(k-p))n+1]bkrk-p)×(1-∑k=p+1∞‍(pp+λ(k-p))n+1akrk-p+∑k=p∞‍(pp+λ(k-p))n+1bkrk-p)-1≥0.
If the condition (18) does not hold, then the numerator in (20) is negative for r sufficiently close to 1. Hence there exists z0=r0 in (0,1) for which the quotient in (20) is negative. This contradicts the required condition for fn∈ℋp,λ-(n;α), and so the proof of Theorem 2 is completed.


## 3. Extreme Points and Distortion Theorem

Our next theorem is on the extreme points of convex hulls of the classℋp,λ-(n;α) denoted by clcoℋp,λ-(n;α).Theorem 3.
Letfn=h+g¯n, where h and gn are given by (6). Then, fn∈ℋp,λ-(n;α) if and only if
(21)fn(z)=∑k=p∞‍[xkhk(z)+ykgnk(z)],
where
(22)h1(z)=zp,hk(z)=zp-1Ψp,λ(n,k,α)zk(k=p+1,p+2,p+3,…),gnk(z)=zp+(-1)n-11Φp,λ(n,k,α)z¯k123325(k=p,p+1,p+2,…),xk,yk≥0,xp=1-∑k=p+1∞‍xk-∑k=p∞‍yk.
In particular, the extreme points of the classℋp,λ-(n;α) are {hk} and {gnk}.Proof.
Suppose that(23)fn(z)=∑k=p∞‍(xkhk(z)+ykgnk(z))=∑k=p∞‍(xk+yk)zp-∑k=p+1∞‍1Ψp,λ(n,k,α)xkzk+(-1)n-1∑k=1∞‍1Φp,λ(n,k,α)ykz¯k.
Then,
(24)∑k=p+1∞‍Ψp,λ(n,k,α)(1Ψp,λ(n,k,α)xk)+∑k=1∞‍Φp,λ(n,k,α)(1Φp,λ(n,k,α)yk)=∑k=p+1∞xk+∑k=p∞‍yk=1-xp≤1,
and so fn∈clcoℋp,λ-(n;α).
Conversely, iffn∈clcoℋp,λ-(n;α), then
(25)ak≤1Ψp,λ(n,k,α),bk≤1Φp,λ(n,k,α).
Set that
(26)xk=Ψp,λ(n,k,α)ak(k=p+1,p+2,p+3,…),yk=Φp,λ(n,k,α)bk(k=p,p+1,p+2,…).
Then note that by Theorem 2, 0≤xk≤1,(k=p+1,p+2,p+3,…), and 0≤yk≤1,(k=p,p+1,p+2,…). We define that xp=1-∑k=p+1∞xk-∑k=p∞yk and note that by Theorem 2, xp≥0. Consequently, we obtain fn(z)=∑k=p∞{xkhk(z)+ykgk(z)} as required.The following theorem gives the distortion bounds for functions in the classℋp,λ-(n;α) which yields a covering result for this class.Theorem 4.
Letfn(z)∈ℋp,λ-(n;α). Then, for |z|=r<1, we have
(27)(1-bp)rp-{Γp,λ(n,α)-Δp,λ(n,α)}rp+1≤|fn(z)|≤(1+bp)rp+{Γp,λ(n,α)-Δp,λ(n,α)bp}rp+1,
where
(28)Γp,λ(n,α)=1-α(p/(p+λ))n-α(p/(p+λ))n+1,Δp,λ(n,α)=1+α(p/(p+λ))n-α(p/(p+λ))n+1.
The result is sharp.Proof.
We only prove the right-hand inequality. The proof for the left-hand inequality is similar and will be omitted. Letfn(z)∈ℋp,λ-(n;α). Taking the absolute value of fn, we have
(29)|fn(z)|≤(1+bp)rp+∑k=p+1∞‍(ak+bk)rk≤(1+bp)rp+∑k=p+1∞‍(ak+bk)rp+1=(1+bp)rp+Γp,λ(n,α)×∑k=p+1∞‍1Γp,λ(n,α)(ak+bk)rp+1≤(1+bp)rp+Γp,λ(n,α)rp+1×∑k=p+1∞‍{Ψp,λ(n,k,α)ak+Φp,λ(n,k,α)bk}≤(1+bp)rp+{Γp,λ(n,k,α)-Δp,λ(n,k,α)bp}rp+1.
The bounds given in Theorem4 for functions fn=h+g¯n,whereh and gnof form (6), also hold for functions of form (2) if the coefficient condition (7) is satisfied. The upper bound given for f∈ℋp,λ-(n;α) is sharp, and the equality occurs for the functions
(30)f(z)=zp+bpz¯p-(1-α(p/(p+λ))n-α(p/(p+λ))n+1-1-α(p/(p+λ))n+α(p/(p+λ))n+1bp)z¯p+1,(31)f(z)=zp+bpz¯p-(1-α(p/(p+λ))n-α(p/(p+λ))n+1-1-α(p/(p+λ))n+α(p/(p+λ))n+1bp)zp+1
showing that the bounds given in Theorem 4 are sharp.Remark 5.
(i) Puttingλ=1 in the previous results, we obtain the results of Cotirla [5].
(ii) Puttingλ=1 in the previous results, we obtain the results of Cotirla [6], when β=0.



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