Research article Special Issues

Some geometric properties of certain meromorphically multivalent functions associated with the first-order differential subordination

  • 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{(1λ)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{(1λ)zpf(z)λpzp+1f(z)} over the class Gn(A,B,0) is obtained.



    Throughout our present discussion, we assume that

    n, pN, 1B<1, B<A  and λ<0. (1.1)

    Let Σn(p) be the class of functions of the form

    f(z)=zp+k=nakzkp (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

    (f1f2)(z)=zp+k=nak,1ak,2zkp=(f1f2)(z),

    where

    fj(z)=zp+k=nak,jzkpΣn(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) (zU), if there exists an analytic function w(z) in U such that

    |w(z)||z|andf(z)=g(w(z))(zU).

    If the function g(z) is univalent in U, then

    f(z)g(z)(zU)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:

    (1λ)zpf(z)λpzp+1f(z)1+Az1+Bz(zU). (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{(1λ)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 Reμ0 and μ0, 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)=z1pf(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

    (1α2)z1pf(z)+α2p1z2pf(z)=g(z)+α2p1zg(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

    (1α1)z1pf(z)+α1p1z2pf(z)=α1α2((1α2)z1pf(z)+α2p1z2pf(z))+(1α1α2)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)zp1)p(1(p1)(AB)m=1Bm1rnmαnm+p1), (2.4)
    Re(f(z)zp1)>p(1(p1)(AB)m=1Bm1αnm+p1), (2.5)
    Re(f(z)zp1)p(1+(p1)(AB)m=1(B)m1rnmαnm+p1) (2.6)

    and

    Re(f(z)zp1)<p(1+(p1)(AB)m=1(B)m1αnm+p1)(B1). (2.7)

    All the bounds are sharp for the function fn(z) given by

    fn(z)=zp+p(p1)(AB)m=1(B)m1znm+p(nm+p)(αnm+p1)(zU). (2.8)

    Proof. It is known that for |ξ|σ (σ<1) that

    |1+Aξ1+Bξ1ABσ21B2σ2|(AB)σ1B2σ2 (2.9)

    and

    1Aσ1BσRe(1+Aξ1+Bξ)1+Aσ1+Bσ. (2.10)

    Let f(z)Qn(A,B,α). Then we can write

    (1α)z1pf(z)+αp1z2pf(z)=p1+Aw(z)1+Bw(z)(zU), (2.11)

    where w(z)=wnzn+wn+1zn+1+ is analytic and |w(z)|<1 for zU. By the Schwarz lemma, we know that |w(z)||z|n (zU). It follows from (2.11) that

    (1α)(p1)αz(1α)(p1)α1f(z)+z(1α)(p1)αf(z)=p(p1)αzp1α1(1+Aw(z)1+Bw(z)),

    which implies that

    (z(1α)(p1)αf(z))=p(p1)αzp1α1(1+Aw(z)1+Bw(z)).

    After integration we arrive at

    f(z)=p(p1)αz(1α)(p1)αz0ξp1α1(1+Aw(ξ)1+Bw(ξ))dξ=p(p1)αzp110tp1α1(1+Aw(tz)1+Bw(tz))dt. (2.12)

    Since

    |w(tz)|tnrn(|z|=r<1; 0t1),

    we get from (2.12) and left-hand inequality in (2.10) that, for |z|=r<1,

    Re(f(z)zp1)p(p1)α10tp1α1(1Atnrn1Btnrn)dt=pp(p1)(AB)m=1Bm1rnmαnm+p1, (2.13)

    and, for zU,

    Re(f(z)zp1)>p(p1)α10tp1α1(1Atn1Btn)dt=pp(p1)(AB)m=1Bm1αnm+p1.

    Similarly, by using (2.12) and the right-hand inequality in (2.10), we have (2.6) and (2.7) (with B1).

    Furthermore, for the function fn(z) given by (2.8), we find that fn(z)An(p),

    fn(z)=pzp1+p(p1)(AB)m=1(B)m1znm+p1αnm+p1 (2.14)

    and

    (1α)z1pfn(z)+αp1z2pfn(z)=p+p(AB)m=1(B)m1znm=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

    (p1)(AB)m=1Bm1αnm+p11, (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)zp1)>0(zU).

    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)1p(p1)(AB)m=1Bm1rnm(nm+p)(αnm+p1), (2.16)
    Re(f(z)zp)1+p(p1)(AB)m=1(B)m1rnm(nm+p)(αnm+p1) (2.17)

    and

    Re(f(z)zp)>1p(p1)(AB)m=1Bm1(nm+p)(αnm+p1). (2.18)

    All of the above bounds are sharp.

    Proof. It is obvious that

    f(z)=z0f(ξ)dξ=z10f(tz)dt=zp10tp1f(tz)(tz)p1dt(zU). (2.19)

    Making use of (2.4) in Theorem 2, it follows from (2.19) that

    Re(f(z)zp)=10tp1Re(f(tz)(tz)p1)dt10tp1(pp(p1)(AB)m=1Bm1(rt)nmαnm+p1)dt=1p(p1)(AB)m=1Bm1rnm(nm+p)(αnm+p1),

    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)p1)p(p1)α10up1α1(1A(utr)n1B(utr)n)du>pp(p1)(AB)m=1Bm1tnmαnm+p1(|z|=r<1; 0<t1).

    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 AB1. Then, for |z|=r<1,

    |f(z)|rp+p(p1)(AB)m=1(B)m1rnm+p(nm+p)(αnm+p1) (2.20)

    and

    |f(z)|<1+p(p1)(AB)m=1(B)m1(nm+p)(αnm+p1). (2.21)

    The above bounds are sharp.

    Proof. Since AB1, it follows from (2.9) that

    |1+Aξ1+Bξ||1ABσ21B2σ2|+(AB)σ1B2σ2=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)p1|p(p1)α10tp1α1|1+Aw(utz)1+Bw(utz)|dtp(p1)α10tp1α1(1+A(utr)n1+B(utr)n)dt (2.23)
    <p(p1)α10tp1α1(1+Auntn1+Buntn)dt. (2.24)

    By noting that

    |f(z)|rp10up1|f(uz)(uz)p1|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)(1B0<1; B0<A0; α0>0).

    If

    p(p1)(A0B0)m=1Bm10(m+p)(α0m+p1)12, (2.25)

    then (fg)(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)>1p(p1)(A0B0)m=1Bm10(m+p)(α0m+p1)12(zU).

    Thus the function g(z)zp has the following Herglotz representation:

    g(z)zp=|x|=1dμ(x)1xz(zU), (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

    z1p(fg)(z)=(z1pf(z))(zpg(z))

    and

    z2p(fg)(z)=(z2pf(z))(zpg(z)).

    Thus

    (1α)z1p(fg)(z)+αp1z2p(fg)(z)=(1α)((z1pf(z))(zpg(z)))+αp1((z2pf(z))(zpg(z)))=h(z)g(z)zp, (2.27)

    where

    h(z)=(1α)z1pf(z)+αp1z2pf(z)p1+Az1+Bz(zU). (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

    (1α)z1p(fg)(z)+αp1z2p(fg)(z)=|x|=1h(xz)dμ(x)p1+Az1+Bz(zU).

    This shows that (fg)(z)Q1(A,B,α). The proof of Theorem 5 is completed.

    Theorem 6. Let

    f(z)=zp+k=nap+kzp+kQn(A,B,α). (2.29)

    Then

    |ap+k|p(p1)(AB)(p+k)(αk+p1)(kn). (2.30)

    The result is sharp for each kn.

    Proof. It is known that, if

    φ(z)=j=1bjzjψ(z)(zU),

    where φ(z) is analytic in U and ψ(z)=z+ is analytic and convex univalent in U, then |bj|1 (jN).

    By using (2.29), we have

    (1α)z1pf(z)+αp1z2pf(z)pp(AB)=1p(p1)(AB)k=n(p+k)(αk+p1)ap+kzkz1+Bz(zU). (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+p1)p(p1)(AB)|ap+k|1(kn),

    which gives (2.30).

    Next we consider the function fk(z) given by

    fk(z)=zp+p(p1)(AB)m=1(B)m1zkm+p(km+p)(αkm+p1)(zU; kn).

    Since

    (1α)z1pfk(z)+αp1z2pfk(z)=p1+Azk1+Bzkp1+Az1+Bz(zU)

    and

    fk(z)=zp+p(p1)(AB)(p+k)(αk+p1)zp+k+

    for each kn, 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{(1α)z1pf(z)+αp1z2pf(z)}p[p1((p1)(A+B)+αn(AB))rn+(p1)ABr2n](p1)(1Brn)2; (2.32)

    (ii) if Mn(A,B,α,r)0, we have

    Re{(1α)z1pf(z)+αp1z2pf(z)}p(4α2KAKBL2n)4α(p1)(AB)rn1(1r2)KB, (2.33)

    where

    {KA=1A2r2nnArn1(1r2),KB=1B2r2nnBrn1(1r2),Ln=2α(1ABr2n)αn(A+B)rn1(1r2)(p1)(AB)rn1(1r2),Mn(A,B,α,r)=2αKB(1Arn)Ln(1Brn). (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)pzp1=1+Aznφ(z)1+Bznφ(z)(zU), (2.35)

    where φ(z) is analytic and |φ(z)|1 in U. It follows from (2.35) that

    (1α)z1pf(z)+αp1z2pf(z)=f(z)zp1+αp(AB)(nznφ(z)+zn+1φ(z))(p1)(1+Bznφ(z))2=f(z)zp1+αnp(p1)(AB)(f(z)pzp11)(ABf(z)pzp1)+αp(AB)zn+1φ(z)(p1)(1+Bznφ(z))2. (2.36)

    By using the Carathéodory inequality:

    |φ(z)|1|φ(z)|21r2,

    we obtain

    Re{zn+1φ(z)(1+Bznφ(z))2}rn+1(1|φ(z)|2)(1r2)|1+Bznφ(z)|2=r2n|ABf(z)pzp1|2|f(z)pzp11|2(AB)2rn1(1r2). (2.37)

    Put f(z)pzp1=u+iv(u,vR). Then (2.36) and (2.37), together, yield

    Re{(1α)z1pf(z)+αp1z2pf(z)}p(1+αn(A+B)(p1)(AB))uαnpA(p1)(AB)αnpB(p1)(AB)(u2v2)αp[r2n((ABu)2+(Bv)2)((u1)2+v2)](p1)(AB)rn1(1r2)=p(1+αn(A+B)(p1)(AB))uαnp(p1)(AB)(A+Bu2)αp(r2n(ABu)2(u1)2)(p1)(AB)rn1(1r2)+αp(p1)(AB)(nB+1B2r2nrn1(1r2))v2. (2.38)

    We note that

    1B2r2nrn1(1r2)1r2nrn1(1r2)=1rn1(1+r2+r4++r2(n2)+r2(n1))=12rn1[(1+r2(n1))+(r2+r2(n2))++(r2(n1)+1)]nnB. (2.39)

    Combining (2.38) and (2.39), we have

    Re{(1α)z1pf(z)+αp1z2pf(z)}p(1+αn(A+B)(p1)(AB))uαnp(p1)(AB)(A+Bu2)+αp((u1)2r2n(ABu)2)(p1)(AB)rn1(1r2)=:ψn(u). (2.40)

    Also, (2.10) and (2.35) imply that

    1Arn1Brnu=Re(f(z)pzp1)1+Arn1+Brn.

    We now calculate the minimum value of ψn(u) on the segment [1Arn1Brn,1+Arn1+Brn]. Obviously, we get

    ψn(u)=p(1+αn(A+B)(p1)(AB))2αnpB(p1)(AB)u+2αp((1B2r2n)u(1ABr2n))(p1)(AB)rn1(1r2),
    ψn(u)=2αp(p1)(AB)(1B2r2nrn1(1r2)nB)2αnp(1B)(p1)(AB)>0(see (2.36)) (2.41)

    and ψn(u)=0 if and only if

    u=un=2α(1ABr2n)αn(A+B)rn1(1r2)(p1)(AB)rn1(1r2)2α(1B2r2nnBrn1(1r2))=Ln2αKB(see (2.31)). (2.42)

    Since

    2αKB(1+Arn)Ln(1+Brn)=2α[(1+Arn)(1B2r2n)(1+Brn)(1ABr2n)]+αnrn1(1r2)[(A+B)(1+Brn)2B(1+Arn)]+(p1)(AB)rn1(1r2)(1+Brn)=2α(AB)rn(1+Brn)+αn(AB)rn1(1r2)(1Brn)+(p1)(AB)rn1(1r2)(1+Brn)>0,

    we see that

    un<1+Arn1+Brn. (2.43)

    But un is not always greater than 1Arn1Brn. The following two cases arise.

    (i) un1Arn1Brn, that is, Mn(A,B,α,r)0 (see (2.34)). In view of ψn(un)=0 and (2.41), the function ψn(u) is increasing on the segment [1Arn1Brn,1+Arn1+Brn]. Therefore, we deduce from (2.40) that, if Mn(A,B,α,r)0, then

    Re{(1α)z1pf(z)+αp1z2pf(z)}ψn(1Arn1Brn)=p(1+αn(A+B)(p1)(AB))(1Arn1Brn)αnp(p1)(AB)(A+B(1Arn1Brn)2)=p1Arn1Brnαnp(p1)(AB)(11Arn1Brn)(AB1Arn1Brn)=p[p1((p1)(A+B)+αn(AB))rn+(p1)ABr2n](p1)(1Brn)2.

    This proves (2.32).

    Next we consider the function f(z) given by

    f(z)=pz0tp11Atn1BtndtQn(A,B,0).

    It is easy to find that

    (1α)r1pf(r)+αp1r2pf(r)=p[p1((p1)(A+B)+αn(AB))rn+(p1)ABr2n](p1)(1Brn)2,

    which shows that the inequality (2.32) is sharp.

    (ii) un1Arn1Brn, that is, Mn(A,B,α,r)0. In this case, we easily see that

    Re{(1α)z1pf(z)+αp1z2pf(z)}ψn(un). (2.44)

    In view of (2.34), ψn(u) in (2.40) can be written as follows:

    ψn(u)=p(αKBu2Lnu+αKA)(p1)(AB)rn1(1r2). (2.45)

    Therefore, if Mn(A,B,α,r)0, then it follows from (2.42), (2.44) and (2.45) that

    Re{(1α)z1pf(z)+αp1z2pf(z)}p(αKBu2nLnun+αKA)(p1)(AB)rn1(1r2)=p(4α2KAKBL2n)4α(p1)(AB)rn1(1r2)KB.

    To show that the inequality (2.33) is sharp, we take

    f(z)=pz0tp11+Atnφ(t)1+Btnφ(t)dtandφ(z)=zcn1cnz(zU),

    where cnR is determined by

    f(r)prp1=1+Arnφ(r)1+Brnφ(r)=un[1Arn1Brn,1+Arn1+Brn).

    Clearly, 1φ(r)<1, 1cn<1, |φ(z)|1 (zU), and so f(z)Qn(A,B,0). Since

    φ(r)=1c2n(1cnr)2=1|φ(r)|21r2,

    from the above argument we obtain that

    (1α)r1pf(r)+αp1r2pf(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{(1λ)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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