A new subclass Gn(A,B,λ) of meromorphically multivalent functions defined by the first-order differential subordination is introduced. Some geometric properties of this new subclass are investigated. The sharp upper bound on |z|=r<1 for the functional Re{(1−λ)zpf(z)−λpzp+1f′(z)} over the class Gn(A,B,0) is obtained.
Citation: Ying Yang, Jin-Lin Liu. Some geometric properties of certain meromorphically multivalent functions associated with the first-order differential subordination[J]. AIMS Mathematics, 2021, 6(4): 4197-4210. doi: 10.3934/math.2021248
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A new subclass Gn(A,B,λ) of meromorphically multivalent functions defined by the first-order differential subordination is introduced. Some geometric properties of this new subclass are investigated. The sharp upper bound on |z|=r<1 for the functional Re{(1−λ)zpf(z)−λpzp+1f′(z)} over the class Gn(A,B,0) is obtained.
Throughout our present discussion, we assume that
n, p∈N, −1≤B<1, B<A and λ<0. | (1.1) |
Let Σn(p) be the class of functions of the form
f(z)=z−p+∞∑k=nakzk−p | (1.2) |
which are analytic in the punctured open unit disk U∗={z:0<|z|<1}. The class Σn(p) is closed under the Hadamard product
(f1∗f2)(z)=z−p+∞∑k=nak,1ak,2zk−p=(f1∗f2)(z), |
where
fj(z)=z−p+∞∑k=nak,jzk−p∈Σn(p)(j=1,2). |
For functions f(z) and g(z) analytic in U={z:|z|<1}, we say that f(z) is subordinate to g(z) and write f(z)≺g(z) (z∈U), if there exists an analytic function w(z) in U such that
|w(z)|≤|z|andf(z)=g(w(z))(z∈U). |
If the function g(z) is univalent in U, then
f(z)≺g(z)(z∈U)⇔f(0)=g(0)andf(U)⊂g(U). |
In this paper we introduce and investigate the following subclass of Σn(p).
Definition. A function f(z)∈Σn(p) is said to be in the class Gn(A,B,λ) if it satisfies the first-order differential subordination:
(1−λ)zpf(z)−λpzp+1f′(z)≺1+Az1+Bz(z∈U). | (1.3) |
Recently, several authors (see, e.g., [1,2,3,4,5,6,7,8,10,11,12,13,14,15,16] and the references cited therein) introduced and studied various subclasses of meromorphically multivalent functions. Certain properties such as distortion bounds, inclusion relations and coefficient estimates are given. In this note we obtain inclusion relation, coefficient estimate and sharp bounds on Re(zpf(z)) for functions f(z) belonging to the class Gn(A,B,λ). Furthermore, we investigate a new problem, that is, to find
max|z|=r<1Re{(1−λ)zpf(z)−λpzp+1f′(z)}, |
where f(z) varies in the class
Gn(A,B,0)={f(z)∈Σn(p):zpf(z)≺1+Az1+Bz}. | (1.4) |
We need the following lemma in order to derive the main results for the class Gn(A,B,λ).
Lemma [9]. Let g(z) be analytic in U and h(z) be analytic and convex univalent in U with h(0)=g(0). If
g(z)+1μzg′(z)≺h(z), |
where Reμ≥0 and μ≠0, then g(z)≺h(z).
Theorem 1. Let 0<α1<α2. Then Qn(A,B,α2)⊂Qn(A,B,α1).
Proof. Suppose that
g(z)=z1−pf′(z) | (2.1) |
for f(z)∈Qn(A,B,α2). Then the function g(z) is analytic in U with g(0)=p. By using (1.3) and (2.1), we have
(1−α2)z1−pf′(z)+α2p−1z2−pf″(z)=g(z)+α2p−1zg′(z)≺p1+Az1+Bz. | (2.2) |
An application of the above Lemma yields
g(z)≺p1+Az1+Bz. | (2.3) |
By noting that 0<α1α2<1 and that the function 1+Az1+Bz is convex univalent in U, it follows from (2.1)–(2.3) that
(1−α1)z1−pf′(z)+α1p−1z2−pf″(z)=α1α2((1−α2)z1−pf′(z)+α2p−1z2−pf″(z))+(1−α1α2)g(z)≺p1+Az1+Bz. |
This shows that f(z)∈Qn(A,B,α1). The proof of Theorem 1 is completed.
Theorem 2. Let f(z)∈Qn(A,B,α). Then, for |z|=r<1,
Re(f′(z)zp−1)≥p(1−(p−1)(A−B)∞∑m=1Bm−1rnmαnm+p−1), | (2.4) |
Re(f′(z)zp−1)>p(1−(p−1)(A−B)∞∑m=1Bm−1αnm+p−1), | (2.5) |
Re(f′(z)zp−1)≤p(1+(p−1)(A−B)∞∑m=1(−B)m−1rnmαnm+p−1) | (2.6) |
and
Re(f′(z)zp−1)<p(1+(p−1)(A−B)∞∑m=1(−B)m−1αnm+p−1)(B≠−1). | (2.7) |
All the bounds are sharp for the function fn(z) given by
fn(z)=zp+p(p−1)(A−B)∞∑m=1(−B)m−1znm+p(nm+p)(αnm+p−1)(z∈U). | (2.8) |
Proof. It is known that for |ξ|≤σ (σ<1) that
|1+Aξ1+Bξ−1−ABσ21−B2σ2|≤(A−B)σ1−B2σ2 | (2.9) |
and
1−Aσ1−Bσ≤Re(1+Aξ1+Bξ)≤1+Aσ1+Bσ. | (2.10) |
Let f(z)∈Qn(A,B,α). Then we can write
(1−α)z1−pf′(z)+αp−1z2−pf″(z)=p1+Aw(z)1+Bw(z)(z∈U), | (2.11) |
where w(z)=wnzn+wn+1zn+1+⋯ is analytic and |w(z)|<1 for z∈U. By the Schwarz lemma, we know that |w(z)|≤|z|n (z∈U). It follows from (2.11) that
(1−α)(p−1)αz(1−α)(p−1)α−1f′(z)+z(1−α)(p−1)αf″(z)=p(p−1)αzp−1α−1(1+Aw(z)1+Bw(z)), |
which implies that
(z(1−α)(p−1)αf′(z))′=p(p−1)αzp−1α−1(1+Aw(z)1+Bw(z)). |
After integration we arrive at
f′(z)=p(p−1)αz−(1−α)(p−1)α∫z0ξp−1α−1(1+Aw(ξ)1+Bw(ξ))dξ=p(p−1)αzp−1∫10tp−1α−1(1+Aw(tz)1+Bw(tz))dt. | (2.12) |
Since
|w(tz)|≤tnrn(|z|=r<1; 0≤t≤1), |
we get from (2.12) and left-hand inequality in (2.10) that, for |z|=r<1,
Re(f′(z)zp−1)≥p(p−1)α∫10tp−1α−1(1−Atnrn1−Btnrn)dt=p−p(p−1)(A−B)∞∑m=1Bm−1rnmαnm+p−1, | (2.13) |
and, for z∈U,
Re(f′(z)zp−1)>p(p−1)α∫10tp−1α−1(1−Atn1−Btn)dt=p−p(p−1)(A−B)∞∑m=1Bm−1αnm+p−1. |
Similarly, by using (2.12) and the right-hand inequality in (2.10), we have (2.6) and (2.7) (with B≠−1).
Furthermore, for the function fn(z) given by (2.8), we find that fn(z)∈An(p),
f′n(z)=pzp−1+p(p−1)(A−B)∞∑m=1(−B)m−1znm+p−1αnm+p−1 | (2.14) |
and
(1−α)z1−pf′n(z)+αp−1z2−pf″n(z)=p+p(A−B)∞∑m=1(−B)m−1znm=p1+Azn1+Bzn. |
Hence fn(z)∈Qn(A,B,α) and, from (2.14), we conclude that the inequalities (2.4) to (2.7) are sharp. The proof of Theorem 2 is completed.
Corollary. Let f(z)∈Qn(A,B,α). If
(p−1)(A−B)∞∑m=1Bm−1αnm+p−1≤1, | (2.15) |
then f(z) is p-valent close-to-convex in U.
Proof. Let f(z)∈Qn(A,B,α) and (2.15) be satisfied. Then, by using (2.5) in Theorem 2, we see that
Re(f′(z)zp−1)>0(z∈U). |
This shows that f(z) is p-valent close-to-convex in U. The proof of the corollary is completed.
Theorem 3. Let f(z)∈Qn(A,B,α). Then, for |z|=r<1,
Re(f(z)zp)≥1−p(p−1)(A−B)∞∑m=1Bm−1rnm(nm+p)(αnm+p−1), | (2.16) |
Re(f(z)zp)≤1+p(p−1)(A−B)∞∑m=1(−B)m−1rnm(nm+p)(αnm+p−1) | (2.17) |
and
Re(f(z)zp)>1−p(p−1)(A−B)∞∑m=1Bm−1(nm+p)(αnm+p−1). | (2.18) |
All of the above bounds are sharp.
Proof. It is obvious that
f(z)=∫z0f′(ξ)dξ=z∫10f′(tz)dt=zp∫10tp−1f′(tz)(tz)p−1dt(z∈U). | (2.19) |
Making use of (2.4) in Theorem 2, it follows from (2.19) that
Re(f(z)zp)=∫10tp−1Re(f′(tz)(tz)p−1)dt≥∫10tp−1(p−p(p−1)(A−B)∞∑m=1Bm−1(rt)nmαnm+p−1)dt=1−p(p−1)(A−B)∞∑m=1Bm−1rnm(nm+p)(αnm+p−1), |
which gives (2.16).
Similarly, we deduce from (2.6) in Theorem 2 and (2.19) that (2.17) holds true.
Also, with the help of (2.13), we find that
Re(f′(tz)(tz)p−1)≥p(p−1)α∫10up−1α−1(1−A(utr)n1−B(utr)n)du>p−p(p−1)(A−B)∞∑m=1Bm−1tnmαnm+p−1(|z|=r<1; 0<t≤1). |
From this and (2.19), we obtain (2.18).
Furthermore, it is easy to see that the inequalities (2.16)–(2.18) are sharp for the function fn(z) given by (2.8). Now the proof of Theorem 3 is completed.
Theorem 4. Let f(z)∈Qn(A,B,α) and AB≤1. Then, for |z|=r<1,
|f(z)|≤rp+p(p−1)(A−B)∞∑m=1(−B)m−1rnm+p(nm+p)(αnm+p−1) | (2.20) |
and
|f(z)|<1+p(p−1)(A−B)∞∑m=1(−B)m−1(nm+p)(αnm+p−1). | (2.21) |
The above bounds are sharp.
Proof. Since AB≤1, it follows from (2.9) that
|1+Aξ1+Bξ|≤|1−ABσ21−B2σ2|+(A−B)σ1−B2σ2=1+Aσ1+Bσ(|ξ|≤σ<1). | (2.22) |
By virtue of (2.12) and (2.22), we have, for |z|=r<1,
|f′(uz)(uz)p−1|≤p(p−1)α∫10tp−1α−1|1+Aw(utz)1+Bw(utz)|dt≤p(p−1)α∫10tp−1α−1(1+A(utr)n1+B(utr)n)dt | (2.23) |
<p(p−1)α∫10tp−1α−1(1+Auntn1+Buntn)dt. | (2.24) |
By noting that
|f(z)|≤rp∫10up−1|f′(uz)(uz)p−1|du, |
we deduce from (2.23) and (2.24) that the desired inequalities hold true.
The bounds in (2.20) and (2.21) are sharp with the extremal function fn(z) given by (2.8). The proof of Theorem 4 is completed.
Theorem 5. Let f(z)∈Q1(A,B,α) and
g(z)∈Q1(A0,B0,α0)(−1≤B0<1; B0<A0; α0>0). |
If
p(p−1)(A0−B0)∞∑m=1Bm−10(m+p)(α0m+p−1)≤12, | (2.25) |
then (f∗g)(z)∈Q1(A,B,α), where the symbol ∗ denotes the familiar Hadamard product of two analytic functions in U.
Proof. Since g(z)∈Q1(A0,B0,α0), we find from the inequality (2.18) in Theorem 3 and (2.25) that
Re(g(z)zp)>1−p(p−1)(A0−B0)∞∑m=1Bm−10(m+p)(α0m+p−1)≥12(z∈U). |
Thus the function g(z)zp has the following Herglotz representation:
g(z)zp=∫|x|=1dμ(x)1−xz(z∈U), | (2.26) |
where μ(x) is a probability measure on the unit circle |x|=1 and ∫|x|=1dμ(x)=1.
For f(z)∈Q1(A,B,α), we have
z1−p(f∗g)′(z)=(z1−pf′(z))∗(z−pg(z)) |
and
z2−p(f∗g)″(z)=(z2−pf″(z))∗(z−pg(z)). |
Thus
(1−α)z1−p(f∗g)′(z)+αp−1z2−p(f∗g)″(z)=(1−α)((z1−pf′(z))∗(z−pg(z)))+αp−1((z2−pf″(z))∗(z−pg(z)))=h(z)∗g(z)zp, | (2.27) |
where
h(z)=(1−α)z1−pf′(z)+αp−1z2−pf″(z)≺p1+Az1+Bz(z∈U). | (2.28) |
In view of the fact that the function 1+Az1+Bz is convex univalent in U, it follows from (2.26) to (2.28) that
(1−α)z1−p(f∗g)′(z)+αp−1z2−p(f∗g)″(z)=∫|x|=1h(xz)dμ(x)≺p1+Az1+Bz(z∈U). |
This shows that (f∗g)(z)∈Q1(A,B,α). The proof of Theorem 5 is completed.
Theorem 6. Let
f(z)=zp+∞∑k=nap+kzp+k∈Qn(A,B,α). | (2.29) |
Then
|ap+k|≤p(p−1)(A−B)(p+k)(αk+p−1)(k≥n). | (2.30) |
The result is sharp for each k≥n.
Proof. It is known that, if
φ(z)=∞∑j=1bjzj≺ψ(z)(z∈U), |
where φ(z) is analytic in U and ψ(z)=z+⋯ is analytic and convex univalent in U, then |bj|≤1 (j∈N).
By using (2.29), we have
(1−α)z1−pf′(z)+αp−1z2−pf″(z)−pp(A−B)=1p(p−1)(A−B)∞∑k=n(p+k)(αk+p−1)ap+kzk≺z1+Bz(z∈U). | (2.31) |
In view of the fact that the function z1+Bz is analytic and convex univalent in U, it follows from (2.31) that
(p+k)(αk+p−1)p(p−1)(A−B)|ap+k|≤1(k≥n), |
which gives (2.30).
Next we consider the function fk(z) given by
fk(z)=zp+p(p−1)(A−B)∞∑m=1(−B)m−1zkm+p(km+p)(αkm+p−1)(z∈U; k≥n). |
Since
(1−α)z1−pf′k(z)+αp−1z2−pf″k(z)=p1+Azk1+Bzk≺p1+Az1+Bz(z∈U) |
and
fk(z)=zp+p(p−1)(A−B)(p+k)(αk+p−1)zp+k+⋯ |
for each k≥n, the proof of Theorem 6 is completed.
Theorem 7. Let f(z)∈Qn(A,B,0). Then, for |z|=r<1,
(i) if Mn(A,B,α,r)≥0, we have
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥p[p−1−((p−1)(A+B)+αn(A−B))rn+(p−1)ABr2n](p−1)(1−Brn)2; | (2.32) |
(ii) if Mn(A,B,α,r)≤0, we have
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥p(4α2KAKB−L2n)4α(p−1)(A−B)rn−1(1−r2)KB, | (2.33) |
where
{KA=1−A2r2n−nArn−1(1−r2),KB=1−B2r2n−nBrn−1(1−r2),Ln=2α(1−ABr2n)−αn(A+B)rn−1(1−r2)−(p−1)(A−B)rn−1(1−r2),Mn(A,B,α,r)=2αKB(1−Arn)−Ln(1−Brn). | (2.34) |
The above results are sharp.
Proof. Equality in (2.32) occurs for z=0. Thus we assume that 0<|z|=r<1.
For f(z)∈Qn(A,B,0), we can write
f′(z)pzp−1=1+Aznφ(z)1+Bznφ(z)(z∈U), | (2.35) |
where φ(z) is analytic and |φ(z)|≤1 in U. It follows from (2.35) that
(1−α)z1−pf′(z)+αp−1z2−pf″(z)=f′(z)zp−1+αp(A−B)(nznφ(z)+zn+1φ′(z))(p−1)(1+Bznφ(z))2=f′(z)zp−1+αnp(p−1)(A−B)(f′(z)pzp−1−1)(A−Bf′(z)pzp−1)+αp(A−B)zn+1φ′(z)(p−1)(1+Bznφ(z))2. | (2.36) |
By using the Carathéodory inequality:
|φ′(z)|≤1−|φ(z)|21−r2, |
we obtain
Re{zn+1φ′(z)(1+Bznφ(z))2}≥−rn+1(1−|φ(z)|2)(1−r2)|1+Bznφ(z)|2=−r2n|A−Bf′(z)pzp−1|2−|f′(z)pzp−1−1|2(A−B)2rn−1(1−r2). | (2.37) |
Put f′(z)pzp−1=u+iv(u,v∈R). Then (2.36) and (2.37), together, yield
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥p(1+αn(A+B)(p−1)(A−B))u−αnpA(p−1)(A−B)−αnpB(p−1)(A−B)(u2−v2)−αp[r2n((A−Bu)2+(Bv)2)−((u−1)2+v2)](p−1)(A−B)rn−1(1−r2)=p(1+αn(A+B)(p−1)(A−B))u−αnp(p−1)(A−B)(A+Bu2)−αp(r2n(A−Bu)2−(u−1)2)(p−1)(A−B)rn−1(1−r2)+αp(p−1)(A−B)(nB+1−B2r2nrn−1(1−r2))v2. | (2.38) |
We note that
1−B2r2nrn−1(1−r2)≥1−r2nrn−1(1−r2)=1rn−1(1+r2+r4+⋯+r2(n−2)+r2(n−1))=12rn−1[(1+r2(n−1))+(r2+r2(n−2))+⋯+(r2(n−1)+1)]≥n≥−nB. | (2.39) |
Combining (2.38) and (2.39), we have
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥p(1+αn(A+B)(p−1)(A−B))u−αnp(p−1)(A−B)(A+Bu2)+αp((u−1)2−r2n(A−Bu)2)(p−1)(A−B)rn−1(1−r2)=:ψn(u). | (2.40) |
Also, (2.10) and (2.35) imply that
1−Arn1−Brn≤u=Re(f′(z)pzp−1)≤1+Arn1+Brn. |
We now calculate the minimum value of ψn(u) on the segment [1−Arn1−Brn,1+Arn1+Brn]. Obviously, we get
ψ′n(u)=p(1+αn(A+B)(p−1)(A−B))−2αnpB(p−1)(A−B)u+2αp((1−B2r2n)u−(1−ABr2n))(p−1)(A−B)rn−1(1−r2), |
ψ″n(u)=2αp(p−1)(A−B)(1−B2r2nrn−1(1−r2)−nB)≥2αnp(1−B)(p−1)(A−B)>0(see (2.36)) | (2.41) |
and ψ′n(u)=0 if and only if
u=un=2α(1−ABr2n)−αn(A+B)rn−1(1−r2)−(p−1)(A−B)rn−1(1−r2)2α(1−B2r2n−nBrn−1(1−r2))=Ln2αKB(see (2.31)). | (2.42) |
Since
2αKB(1+Arn)−Ln(1+Brn)=2α[(1+Arn)(1−B2r2n)−(1+Brn)(1−ABr2n)]+αnrn−1(1−r2)[(A+B)(1+Brn)−2B(1+Arn)]+(p−1)(A−B)rn−1(1−r2)(1+Brn)=2α(A−B)rn(1+Brn)+αn(A−B)rn−1(1−r2)(1−Brn)+(p−1)(A−B)rn−1(1−r2)(1+Brn)>0, |
we see that
un<1+Arn1+Brn. | (2.43) |
But un is not always greater than 1−Arn1−Brn. The following two cases arise.
(i) un≤1−Arn1−Brn, that is, Mn(A,B,α,r)≥0 (see (2.34)). In view of ψ′n(un)=0 and (2.41), the function ψn(u) is increasing on the segment [1−Arn1−Brn,1+Arn1+Brn]. Therefore, we deduce from (2.40) that, if Mn(A,B,α,r)≥0, then
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥ψn(1−Arn1−Brn)=p(1+αn(A+B)(p−1)(A−B))(1−Arn1−Brn)−αnp(p−1)(A−B)(A+B(1−Arn1−Brn)2)=p1−Arn1−Brn−αnp(p−1)(A−B)(1−1−Arn1−Brn)(A−B1−Arn1−Brn)=p[p−1−((p−1)(A+B)+αn(A−B))rn+(p−1)ABr2n](p−1)(1−Brn)2. |
This proves (2.32).
Next we consider the function f(z) given by
f(z)=p∫z0tp−11−Atn1−Btndt∈Qn(A,B,0). |
It is easy to find that
(1−α)r1−pf′(r)+αp−1r2−pf″(r)=p[p−1−((p−1)(A+B)+αn(A−B))rn+(p−1)ABr2n](p−1)(1−Brn)2, |
which shows that the inequality (2.32) is sharp.
(ii) un≥1−Arn1−Brn, that is, Mn(A,B,α,r)≤0. In this case, we easily see that
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥ψn(un). | (2.44) |
In view of (2.34), ψn(u) in (2.40) can be written as follows:
ψn(u)=p(αKBu2−Lnu+αKA)(p−1)(A−B)rn−1(1−r2). | (2.45) |
Therefore, if Mn(A,B,α,r)≤0, then it follows from (2.42), (2.44) and (2.45) that
Re{(1−α)z1−pf′(z)+αp−1z2−pf″(z)}≥p(αKBu2n−Lnun+αKA)(p−1)(A−B)rn−1(1−r2)=p(4α2KAKB−L2n)4α(p−1)(A−B)rn−1(1−r2)KB. |
To show that the inequality (2.33) is sharp, we take
f(z)=p∫z0tp−11+Atnφ(t)1+Btnφ(t)dtandφ(z)=−z−cn1−cnz(z∈U), |
where cn∈R is determined by
f′(r)prp−1=1+Arnφ(r)1+Brnφ(r)=un∈[1−Arn1−Brn,1+Arn1+Brn). |
Clearly, −1≤φ(r)<1, −1≤cn<1, |φ(z)|≤1 (z∈U), and so f(z)∈Qn(A,B,0). Since
φ′(r)=−1−c2n(1−cnr)2=−1−|φ(r)|21−r2, |
from the above argument we obtain that
(1−α)r1−pf′(r)+αp−1r2−pf″(r)=ψn(un). |
The proof of Theorem 7 is completed.
In this paper, we have introduced and investigated some geometric properties of the class Gn(A,B,λ) which is defined by using the principle of first-order differential subordination. For this function class, we have derived the sharp upper bound on |z|=r<1 for the following functional:
Re{(1−λ)zpf(z)−λpzp+1f′(z)} |
over the class Gn(A,B,0). We have also obtained other properties of the function class Gn(A,B,λ).
Motivated by a recently-published survey-cum-expository review article by Srivastava [15], the interested reader's attention is drawn toward the possibility of investigating the basic (or q-) extensions of the results which are presented in this paper. However, as already pointed out by Srivastava, their further extensions using the so-called (p, q)-calculus will be rather trivial and inconsequential variations of the suggested extensions which are based upon the classical q-calculus, the additional paremeter p being redundant or superfluous (see, for details, [15, p. 340]).
The authors would like to express sincere thanks to the referees for careful reading and suggestions which helped us to improve the paper. This work was supported by National Natural Science Foundation of China (Grant No.11571299).
The authors agree with the contents of the manuscript, and there are no conflicts of interest among the authors.
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