Abstract
In this work, sufficient conditions on the sequence \(\{a_n\}\) are obtained to guarantee the starlikeness, close-to-convexity and convex in the direction of imaginary axis of the analytic function \(f(z)=z+\displaystyle \sum \nolimits _{n=2}^{\infty }a_nz^n\) in the unit disc \(\mathbb {D}.\) These results are obtained using the positivity technique of trigonometric sum as a tool. These coefficient conditions are extended to the triplet \((a,\,b,\,c)\) to ensure that the normalized Gaussian hypergeometric function \(zF(a,\,b;\,c;\,z)\) is starlike. Examples are provided to compare the obtained conditions with the existing results in the literature.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction
Let \(\mathbb {D}=\{z{\text {:}}\,|z|<1\}.\) For \(a,\,b,\,c\in \mathbb {C}\) and \(c\ne 0,\,-1,\,-2,\ldots ,\) the Gaussian hypergeometric function is defined by
where \((a)_k\) is the Pochhammer symbol defined by
k is a positive integer and \(\varGamma \) denotes the Gamma function. Finding the condition on the triplet \((a,\,b,\,c)\) so that the normalized Gaussian hypergeometric function \(zF(a,\,b;\,c;\,z)\) is starlike and convex in \(\mathbb {D}\) has been discussed by several authors, but the complete range on the triplet \((a,\,b,\,c)\) for this problem is still open. For example, see [6, 9] and the references therein.
In the given context, a function is starlike if it maps \(\mathbb {D}\) onto a domain starlike with respect to origin and convex if it maps \(\mathbb {D}\) onto a convex domain. \(\mathcal {S}^{*}\) and \(\mathcal {C}\) denote, respectively, the class of all starlike functions and convex functions in \(\mathbb {D}.\) These two classes are well known subclasses of the class \(\mathcal {S}\) of univalent (one-to-one) functions in \(\mathbb {D},\) which itself is contained in the class \(\mathcal {A}\) of analytic functions f, with the normalization \(f(0)=f^{\prime }(0)-1=0.\) Naturally, \(\mathcal {S}\) and its subclasses are also assumed to have this normalization.
These two subclasses \(\mathcal {S}^{*}\) and \(\mathcal {C}\) also have their respective generalizations \(\mathcal {S}^{*}(\alpha )\) and \(\mathcal {C}(\alpha ),\,0\le \alpha <1,\) as class of starlike function of order \(\alpha \) and class of convex function of order \(\alpha ,\) respectively, with the analytic characterization.
\(0\le \alpha <1\) and \(z\in \mathbb {D}.\) Note that \(\mathcal {S}^{*}(0)=\mathcal {S}^{*}\) and \(\mathcal {C}(0)=\mathcal {C}.\) The relation between these two classes is given by famous Alexander transform,
Another important subclass of \(\mathcal {S}\) is the class of close-to-convex functions of order \(\alpha ,\, \alpha \in [0,\,1),\) with respect to a starlike function \(g\in \mathcal {S}^{*}\) given by the analytic characterization,
In the sequel, we consider only its particular case where \(\gamma =0.\) Several interesting properties of this and other subclasses can be found in [4, 5, 14] and the references therein.
A function \(f\in \mathcal {A}\) satisfying \(\mathrm{Im}(z)\mathrm{Im}(f(z))>0\) where \(z\in \mathbb {D}\) is said to be typically real function. A function \(f\in \mathcal {A}\) is convex in the direction of imaginary axis if every line parallel to imaginary axis either intersect \(f(\mathbb {D})\) in a line segment or has an empty intersection. In [15] it is proved that \(f\in \mathcal {A}\) with real coefficient is convex in the direction of imaginary axis if \(zf^{\prime }\) is typically real which is equivalent to \(\mathrm{Re}[(1-z^2)f^{\prime }(z)]>0.\)
Ruscheweyh [16] proved that if for \(f\in \mathcal {A},\,f^{\prime }\) is typically real and \(\mathrm{Re}(f^{\prime }(z))>0\) in \(\mathbb {D}\) then f is necessarily starlike and this result was extended to starlike of order \(\alpha \) in [12]. We state the extension as a lemma.
Lemma 1.1
([12], Theorem 3.1) Let \(0\le \alpha <1\) and \(f\in \mathcal {A}\) be such that \(f^{\prime }(z)\) and \(f^{\prime }(z)-\alpha \frac{f(z)}{z}\) are typically real in \(\mathbb {D}.\) Further if \(\mathrm{Re}f^{\prime }(z)>0\) and \(\mathrm{Re}\left( f^{\prime }(z)-\alpha \frac{f(z)}{z}\right) >0,\) then \(f\in \mathcal {S}^{*}(\alpha ).\)
Various techniques are used to obtain the geometric and analytic properties of these subclasses of \(\mathcal {S}.\) Among these, the positivity of trigonometric sum is of recent interest. One of the remarkable results for the positivity of sine and cosine sum is given by Vietoris [17].
Theorem 1.1
[17] If \(a_0\ge a_1 \ge a_2 \cdots \ge a_n >0\) and \(2k a_{2k}\le (2k-1) a_{2k-1},\,k\ge 1,\) then for \(n\ge 1\) and \(\theta \in (0,\,\pi ),\)
Among various results involving the coefficients obtained in [17], the inequalities for the specific case in which \(a_k=c_k\) is important with reference to this work, where the sequence \(c_k\) is defined as
Theorem 1.1 has several interesting applications in geometric function theory. Using Theorem 1.1, Ruscheweyh [16] obtained coefficient conditions for \(f\!\in \!\mathcal {A}\) to be starlike.
Theorem 1.2
[16] Let \(a_1=1\) and \(a_k\ge 0\) satisfy
Then \(f(z)=\displaystyle \sum \nolimits _{k=1}^{\infty }a_kz^k\) is starlike.
Among several generalizations of Theorem 1.1, we are interested in the following result given in [13].
Theorem 1.3
[13] Let \(\beta \ge 0,\,\mu \in \mathbb {R}\) such that \(0<\mu +\beta <1\) and \(n\in \mathbb {N}.\) If \(d_0=d_1=1\) and \(d_{2k}=d_{2k+1}=\frac{(1+\beta )_{n-k}n!}{(n-k)!(1+\beta )_n}. \frac{(\mu +\beta )_k}{k!}\) for \(1\le k\le n.\) Then
-
(1)
\(\displaystyle \sum \nolimits _{k=0}^{n}d_k\cos k\theta >0 \Longleftrightarrow \mu +\beta \le \mu ^{*}(1/2)=0.691556,\ldots ,\)
-
(2)
\(\displaystyle \sum \nolimits _{k=1}^{2n+1}d_k\sin k\theta >0 \Longleftrightarrow \mu +\beta \le \mu ^{*}(1/2),\)
-
(3)
\(\displaystyle \sum \nolimits _{k=1}^{2n}d_k\sin k\theta >0\) for \(\mu +\beta \le \dfrac{1+\beta }{2},\)
where \(\mu ^{*}(\lambda ),\,\lambda \in (0,\,1]\) is the unique solution in \((0,\,1)\) of
Note that this \(\mu ^{*}(\lambda )\) was first obtained by Koumandos and Ruscheweyh [8]. The particular case \(\mu ^{*}(1/2)=\mu _0^{*}\) is used in this work extensively.
This work is organized as follows: in the next section, certain conditions on the starlikeness of \(f\in \mathcal {A}\) by means of its Taylor coefficients are obtained using Theorem 1.3. Conditions on the coefficient \(\{a_k\}\) are obtained that ensure that the function \(f(z)=\sum _{k=1}^{\infty }a_kz^k\) is starlike and also the partial sum \(f_n(z)\) to be close-to-convex or convex in the direction of imaginary axis. In the last section, conditions on the triplet \((a,\,b,\,c)\) are obtained for which the normalized Gaussian hypergeometric function \(zF(a,\,b;\,c;\,z)\) is starlike of certain order.
2 Main Results
The positivity of the cosine sum and sine sum given in Theorem 1.3 [13] can be improved as given below by direct application of Abel’s summation formula.
Lemma 2.1
Let \(\beta \ge 0,\,\mu \in \mathbb {R}\) such that \(0<\mu +\beta <1\) and \(n\in \mathbb {N}.\) If \(\{a_k\}\) be decreasing sequence of non- negative numbers satisfying \(a_0>0\) and
then for all \(0<\theta <\pi ,\)
Proof
Let the sequence \(\{d_k\}\) be as given in Theorem 1.3. Then Theorem 1.3 guarantees that \(\displaystyle \sum \nolimits _{k=0}^{n}d_k\cos k\theta >0\) by the hypothesis given in the statement. Using summation by parts, if \(\displaystyle \sum \nolimits _{k=0}^{n}a_k\cos k\theta \) is rewritten as
then
which gives the desired result. \(\square \)
Following the same procedure as in Lemma 2.1 the following result for sine sums can be obtained.
Lemma 2.2
Let \(0\le \beta \le 2\mu _0^{*}-1,\, \mu \in \mathbb {R}\) such that \(0<\mu +\beta <1\) and \(n\in \mathbb {N}.\) If \(\{a_k\}\) be decreasing sequence of non-negative numbers satisfying \(a_0>0\) and (2.1), then for all \(0<\theta <\pi ,\)
The above two lemmas can be used to find a geometric property of an arbitrary but normalized analytic function, using the positivity behaviour related to its Taylor coefficients.
Theorem 2.1
Let \(0\le \beta \le 2\mu _0^{*}-1\) and \(-\beta <\mu \le \frac{1-\beta }{2},\, a_1=1,\,a_k\ge 0\) satisfy
for \(1\le k \le n.\) Then \(f_n(z)=\displaystyle \sum \nolimits _{k=1}^{n}a_kz^k\) is starlike of order \(\frac{1-2\mu -\beta }{(1+\beta )(1-\mu -\beta )}.\) Moreover in the limiting case, \(f(z)=\displaystyle \lim \nolimits _{n\rightarrow \infty }f_n(z)=\displaystyle \sum \nolimits _{k=1}^{\infty }a_kz^k\) is starlike of the same order if \(\{a_k\}\) satisfy (2.2) and in addition
Proof
Let \(f_{n}(z)=z+\displaystyle \sum \nolimits _{k=2}^{n}a_kz^k\) and \(\gamma :=\dfrac{1-2\mu -\beta }{(1+\beta )(1-\mu -\beta )}.\) Then
where \(b_0=(1-\gamma )\) and \(b_k=(k+1-\gamma )a_{k+1}\) for \(1\le k\le n-1.\) Clearly \(b_0>0.\) Further, we claim that these \(\{b_k\}\) are decreasing and satisfy (2.1) for \(0\le k\le n-1.\) To prove that the sequence \(\{b_k\}\) is decreasing it is enough to establish that \((k-\gamma )a_k\ge (k+1-\gamma )a_{k+1}\) or equivalently
which by rearranging the terms, leads to (2.2). To verify that \(\{b_k\}\) satisfy (2.1), we write
which is nothing but (2.3) given in the hypothesis.
Thus \(\{b_k\}\) satisfy the hypothesis of Lemmas 2.1 and 2.2. Hence for \(0<\mu +\beta \le (1+\beta )/2,\, \displaystyle \sum \nolimits _{k=0}^{n}b_k\cos k\theta >0\) and \(\displaystyle \sum \nolimits _{k=1}^{n}b_k\sin k\theta >0\) for \(\theta \in (0,\,\pi ).\) By the minimum principle of harmonic functions we obtain that
Moreover \(\mathrm {Im}(g_n(z))=\displaystyle \sum \nolimits _{k=1}^{n}b_kr^k\sin k\theta \equiv 0\) if \(-1<z=x+iy<1\) and \(\mathrm {Im}(g_n(z))>0\) in \(\mathbb {D}\cap \{z{\text {:}}\,\mathrm {Im}(z)>0\}.\) The reflection principle yields \(\mathrm {Im}(g_n(z))<0\) in \(\mathbb {D}\cap \{z{\text {:}}\,\mathrm {Im}(z)<0\}.\) Thus \(g_n(z)\) is typically real.
Now to complete the proof of the theorem, it remains to justify that \(f^{\prime }_n(z)\) is typically real having real part. Now,
By the same argument as above, we have that \(\mathrm {Re}(f^{\prime }_n(z))>0\) and typically real function. So by Lemma 1.1, we get that \(f_n(z)\) is starlike of order \(\dfrac{1-2\mu -\beta }{(1+\beta )(1-\mu -\beta )}.\) Clearly in the limiting case as \(n\rightarrow \infty \) (2.3) becomes (2.4) and since the family of starlike functions of a fixed order is normal, we get \(f(z)=\displaystyle \lim \nolimits _{n\rightarrow \infty }f_n(z)\) is starlike of order \(\dfrac{1-2\mu -\beta }{(1+\beta )(1-\mu -\beta )}\) if \(\{a_k\}\) satisfy (2.2) and (2.4). \(\square \)
Remark 2.1
By using \(\beta =2\mu _0^{*}-1\) and \(\mu =1-\mu _0^{*}\) in Theorem 2.1, inequalities (2.2) and (2.4) reduce to
Clearly, these inequalities provide a better lower bound for the ratio \(\frac{a_{2k}}{a_{2k+1_{}}}\) compared to the one given in Theorem 1.2. This is due to the fact that, Theorem 1.2 is obtained using Theorem 1.1, whereas Theorem 2.1 is obtained using Theorem 1.3.
Using Lemma 2.1 and minimum principle for harmonic functions, the following result for \(f(z)=z+\displaystyle \sum \nolimits _{k=2}^na_kz^k\) to be close-to-convex with respect to starlike function \(g(z)=z\) is obtained.
Theorem 2.2
Let \(\mu \in \mathbb {R}\) and \(\beta \ge 0\) such that \(0<\mu +\beta <1\) and let \(a_1=1\) and \(a_k\ge 0\) satisfy
and
Then \(f_n(z)=z+\displaystyle \sum \nolimits _{k=2}^n a_k z^k\) satisfies \(\mathrm{Re}(f_n^{\prime }(z))>1-\frac{\mu +\beta }{\mu _0^{*}}.\)
Proof
Let \(\delta =1-\frac{\mu +\beta }{\mu _0^{*}}\) and \(f_n(z) =z+\displaystyle \sum \nolimits _{k=2}^n a_k z^k.\) Then
where \(b_0=1\) and \(b_k=\frac{(k+1)a_{k+1}}{(1-\delta )}\) for \(1\le k\le n-1.\) Clearly \(a_k\ge 0\) implies \(b_k\ge 0\) for \(k\ge 1.\) To prove the theorem it is required that \(\{b_k\}\) are decreasing and satisfy (2.1). Now \(\{b_k\}\) are decreasing if \(\{a_k\}\) satisfy
and \(b_1\le b_0\) implies \(2a_2\le 1-\delta \) which is nothing but (2.5). Moreover \(\{b_k\}\) satisfy (2.1) if \(\{a_k\}\) satisfy
which implies (2.6). Similar arguments using minimum principle for harmonic function yield the required result. \(\square \)
The next result gives the condition for which \(f_n(z)\) is convex in the direction of imaginary axis which is equivalent to the condition that \(f_n(z)\) is close-to-convex with respect to the starlike function \(g(z)=z/(1-z^2).\)
Theorem 2.3
Let \(0\le \beta \le 2\mu _0^{*}-1,\,a_1=1,\,a_k\ge 0\) satisfy
for \(k\ge 1,\,-\beta <\mu \le \frac{1-\beta }{2}.\) Then \(f_n(z)=\displaystyle \sum \nolimits _{k=1}^{n}a_kz^k\) is convex in the direction of imaginary axis.
Proof
\(f_n(z)\) is convex in the direction of imaginary axis which is equivalent to \(zf_n^{\prime }(z)\) that is typically a real function. Also \(f_n(z)\) has real coefficients.
where \(b_k=ka_k.\) To obtain the result, it is required that \(\{b_k\}\) satisfy the conditions given in Lemma 2.2. Then for \(1\le k\le n,\)
and for \(k\in \{1,\,2\ldots ,[n/2]\},\)
Thus \(\{b_k\}\) satisfy the conditions of Lemma 2.2. Then for \(\mu +\beta \in \left( 0,\,\frac{1+\beta }{2}\right] ,\) using minimum principle for harmonic functions \(\mathrm{Im}(zf_n^{\prime }(z))=\displaystyle \sum \nolimits _{k=1}^nb_kr^k\sin {k\theta }>0\) where \(\theta \in (0,\,\pi )\) and \(r\in (0,\,1)\) and \(\mathrm{Im}(zf_n^{\prime }(z))\equiv 0\) for \(z\in \mathbb {D}\cap \{z=x+iy{\text {:}}\,-1<x<1,\,y=0\}.\) Schwarz reflection principle yields that \(\mathrm{Im}(zf_n^{\prime }(z))<0\) for \(\theta \in (\pi ,\,2\pi ).\) So \(zf_n^{\prime }(z)\) is typically real function or equivalently \(f_n(z)\) is convex in the direction of imaginary axis. \(\square \)
Theorem 2.3 generalizes various earlier known results. For example, \(\beta =0\) gives the following corollary due to Ali et al. [3], which for \(\mu =1/2,\) further reduces to a result obtained by Acharya [1].
Corollary 2.1
[3] Let \(\{a_k\}\) be a sequence of non-negative real numbers such that \(a_1=1\) and if it satisfies (2.7) and
then the function \(\displaystyle \sum \nolimits _{k=1}^na_kz^k\) is convex in the direction of imaginary axis whenever \(\mu \in (0,\,1/2].\)
Note that, Theorem 2.3 has a lower bound for \(\frac{a_{2k}}{a_{2k-1}},\) which is lesser than the lower bound given in [11, Theorem 2.3]. The following example supports this observation. However both these results are not sharp.
Example 2.1
Consider the sequence \(\{a_k\}\) such that \(a_1=1\) and
Then by Theorem 2.3 we conclude that \(f(z)=z+\displaystyle \sum \nolimits _{k=2}^na_kz^k\) is convex in the direction of imaginary axis.
3 Starlikeness of Gaussian Hypergeometric Function
Among various special functions, Gaussian hypergeometric functions play an important role in the study of univalent function. Hence the starlikeness of Gaussian hypergeometric function is of considerable interest. Note that in [9], the order of starlikeness is analyzed for various ranges of the parameters \(a,\,b,\,c\) of \(zF(a,\,b;\,c;\,z).\) The starlikeness of \(zF(a,\,b;\,c;\,z)\) using duality technique was studied by several authors including [2]. For other results in this direction, we refer the [3, 6, 7, 10] and the references therein. In this section, we use Theorem 2.1 in finding the starlikeness of the Gaussian hypergeometric function \(zF(a,\,b;\,c;\,z).\) Examples are also given to compare with the range available in the literature. Sufficient conditions for odd Gaussian hypergeometric function \(zF(a,\,b;\,c;\,z^2)\) to be starlike are also obtained.
Theorem 3.1
Let \(0\le \beta \le 2\mu _0^{*}-1\) and \(-\beta <\mu \le \frac{1-\beta }{2}\) and \(a,\,b\le \mu +\beta -1\) satisfy \((a)_k(b)_k\ge 0\) for \(k\ge 2.\) If \(Mc\ge ab\) where \(M=\frac{\beta (1-\mu -\beta )+\mu }{2\beta (1-\mu -\beta )+1-\beta }>0,\) then \(zF(a,\,b;\,c;\,z)\) is starlike of order \(\frac{1-2\mu -\beta }{(1+\beta )(1-\mu -\beta )}.\)
Proof
Clearly \(z F(a,\,b;\,c;\,z) =\displaystyle \sum \nolimits _{k=1}^{\infty }b_k z^k\) implies \(b_1=1\) and \(b_k=\frac{(a)_{k-1}(b)_{k-1}}{(c)_{k-1}(k-1)!}\) for \(k\ge 2.\) Note that \(b_k\) and \(b_{k+1}\) can be related by
To establish the theorem it is enough to prove that \(\{b_k\}\) satisfy the conditions (2.2) and (2.4) of Theorem 2.1. Clearly, a simple computation using the above relation between \(b_k\) and \(b_{k+1}\) provides.
where h(k) is defined as
Let
such that
The conditions \(a\le \mu +\beta -1\le \frac{\beta -1}{2},\, b\le \mu +\beta -1\le \frac{\beta -1}{2}\) and \(Mc \ge ab>0,\) where
show that \(A_1\ge 0,\,A_3\ge 0.\) Note that M can be written as \(M=\frac{(1-\beta )(\mu +\beta )}{2\beta (1-\mu -\beta )+1-\beta }\) so that \(M\ge 0.\) Since \(2\beta (1-\mu -\beta )+1-\beta \ge 0,\) we consider,
Now, using \(Mc\ge ab>0,\) we get
which gives
This means
Thus \(A_1\ge 0,\,A_2\ge 0\) and \(A_3\ge 0\) contribute to \(h(k)\ge 0\) for \(k\ge 1.\)
It remains to verify (2.4)
Clearly,
where
Let
so that \(g(k)=B_1(k-1)^2+B_2(k-1)+B_3.\) We claim that \(B_1,\,B_2,\,B_3\) are non-negative.
For \(a\le \mu +\beta -1\) and \(b\le \mu +\beta -1,\) we have
and
where \(\mathcal {N}:=\mathcal {N}(\mu +\beta )\) is a second-degree polynomial in \((\mu +\beta )\) and \(0<\mu +\beta \le \frac{1+\beta }{2}.\)
Now
Clearly \(\mathcal {N}\) has no zeros in \(\left( 0,\,\frac{1+\beta }{2}\right] .\) So \(\mathcal {N}>0\) implies \(B_2>0.\) \(B_3\) can also be rewritten as
where
To prove our claim it remains to prove that \(B_{31}>0.\) However,
This means \(B_3>0.\) Hence inequality (2.4) holds good. The desired result follows from Theorem 2.1. \(\square \)
Note that Küstner [9] obtained the order of starlikeness of normalize Gaussian hypergeometric function \(z\rightarrow zF(a,\,b;\,c;\,z)\) for all parameters \(a,\,b,\,c\in \mathbb {R},\, c>0\) except for the following ranges:
-
(i)
\(a<-1,\,0<b<c<b-a+1,\)
-
(ii)
\(0<a<c<b-a+1,\,c+1<b,\)
-
(iii)
\(a\le b<0<c,\)
-
(iv)
\(0<c<a\le b,\)
-
(v)
\(a<0<c<b.\)
Theorem 3.1 gives the answer for the range (iii). Since a and b are negative real numbers in Theorem 3.1, and have no relation between them, our results cover the case \(b\le a\) also.
The result in Theorem 3.1 is not sharp. However, this result is better than the earlier results available in the literature for certain ranges of \(a,\,b\) and c. If we take \(\mu =1-\mu _0^{*}\) and \(\beta =2\mu _0^{*}-1\) Theorem 3.1 leads to the following example.
Example 3.1
Let \(a,\,b\le \mu _0^{*}-1\) satisfy \((a)_k(b)_k\ge 0\) for \(k\ge 2\) and if \(c\ge 2ab,\) then \(z F(a,\,b;\,c;\,z)\) is starlike.
Remark 3.1
If we substitute \(\mu +\beta =\frac{1+\beta }{3-\beta }\) in Theorem 3.1, then \(M=1/3\) and \(c\ge 3ab,\) which lead to constructing the following case.
If \(a,\,b\in \left( -1,\,\frac{2(\mu _0^{*}-1)}{2-\mu _0^{*}}\right] \) then \(zF(a,\,b;\,c;\,z)\) is starlike of order 1/2. Note that for the given range of a and b, we get \(c\ge 3ab\) which is better than the range \(c\ge 1+2ab\) provided by Hästö et al. [6, Corollary 1.7]. However, Corollary 1.7 of [6] is valid for all \(a,\,b\) non-zero real, which is not applicable in our case.
Using Alexander transform and
the following consequence of Theorem 3.1 can be deduced which is of considerable interest.
Corollary 3.1
Let \(0\le \beta < 2\mu _0^{*}-1\) and \(-\beta <\mu \le \frac{1-\beta }{2}\) and \(a,\,b\le \mu +\beta -2\) satisfy \((a+1)_k(b+1)_k\ge 0\) for \(k\ge 2.\) If \(M(c+1)\ge (a+1)(b+1),\) then the function \(\frac{c}{ab}[F(a,\,b;\,c,\,z)-1]\) is convex of order \((1-2\mu -\beta )/(1+\beta )(1-\mu -\beta ).\)
Proof
If we replace \(a,\,b,\,c\) by \(a+1,\,b+1,\,c+1\), respectively, in Theorem 3.1 then we get that for \(a,\,b\le \mu +\beta -2,\,(a+1)_k(b+1)_k\ge 0\) for \(k\ge 2\) and if \(M(c+1)\ge (a+1)(b+1)\) then \(F(a+1,\,b+1;\,c+1;\,z)\) is starlike of order \((1-2\mu -\beta )/(1+\beta )(1-\mu -\beta ).\) Since
by Alexander transformation, \(\int _0^z F(a+1,\,b+1;\,c+1;\,t)\frac{\mathrm{d}t}{t}\) is convex of order \((1-2\mu -\beta )/(1+\beta )(1-\mu -\beta )\) for the same conditions. \(\square \)
The next result gives the condition on the triplet \((a,\,b,\,c)\) for the starlikeness of odd Gaussian hypergeometric function.
Theorem 3.2
Let \(0\le \beta < 2\mu _0^{*}-1\) and \(\mu \in \left( -\beta ,\,\frac{(1-\beta )^2}{(3-\beta )}\right] \) and \(a,\,b\le \mu +\beta -1\) satisfy \((a)_k(b)_k\ge 0\) for \(k\ge 2.\) If \(Mc\ge ab,\) then the function \(z F(a,\,b;\,c,\,z^2)\) is starlike of order \(\frac{(1+\beta -(3+\beta )(\mu +\beta ))}{(1+\beta )(1-\mu -\beta )}.\)
Proof
Let \(f(z)=z F(a,\,b;\,c,\,z)\) and \(zg(z)=f(z^2).\) Then \(g(z)=z F(a,\,b;\,c;\,z^2)\) and from Theorem 3.1 \(f\in \mathcal {S}^{*}\left( \frac{1-2\mu -\beta }{(1+\beta )(1-\mu -\beta )}\right) .\) Taking logarithmic derivative of g(z) and considering the real part, we get
\(\Longrightarrow g\in \mathcal {S}^{*}\left( \frac{(1+\beta )-(3+\beta )(\mu +\beta )}{(1+\beta )(1-\mu -\beta )} \right) .\) \(\square \)
For \(\beta =2\mu _0^{*}-1\) and \(\mu +\beta =\frac{1+\beta }{3-\beta },\) we obtain the following corollary of Theorem 3.2.
Corollary 3.2
Let \(a,\,b\le \frac{2(\mu _0^{*}-1)}{2-\mu _0^{*}}\) satisfy \((a)_k(b)_k\ge 0,\,k\ge 2.\) Then for \(c\ge 3ab,\,z F(a,\,b;\,c;\,z^2)\in \mathcal {S}^{*}.\)
Remark 3.2
-
(i)
Corollary 3.2 is better than the result [3, Corollary 3.10] that states that \(z F(a,\,b;\,c;\,z^2)\) is starlike provided \(a,\,b\le -2/3\) and \(c\ge 3ab.\) Corollary 3.2 gives a better range for a and b.
-
(ii)
When \(a,\,b\in \left( -1,\, \frac{2(\mu _0^{*}-1)}{2-\mu _0^{*}}\right] \) then \(zF(a,\,b;\,c;\,z^2)\in \mathcal {S}^{*}\) for \(c\ge 3ab\) which is better than the condition \(c\ge 1+2ab\) provided in [6], but for a smaller range of a and b.
References
Acharya, A.P.: Univalence criteria for analytic functions and applications to hypergeometric functions. PhD Thesis, University of Würzburg (1997)
Ali, R.M., Badghaish, A.O., Ravichandran, V., Swaminathan, A.: Starlikeness of integral transforms and duality. J. Math. Anal. Appl. 385(2), 808–822 (2012)
Ali, R.M., Lee, S.K.,Mondal, S.R.: Coefficient conditions for starlikeness of nonnegative order. Abstr. Appl. Anal. (2012). doi:10.1155/2012/450318
Duren, P.L.: Univalent Functions. Springer, Berlin (1983)
Goodman, A.W.: Univalent Functions, vol. I. Mariner, Tampa (1983)
Hästö, P., Ponnusamy, S., Vuorinen, M.: Starlikeness of the Gaussian hypergeometric functions. Complex Var. Elliptic Equ. 55(1–3), 173–184 (2010)
Kim, J.A., Cho, N.E.: Properties of convolutions for hypergeometric series with univalent functions. Adv. Differ. Equ. 2013, 101 (2013)
Koumandos, S., Ruscheweyh, S.: On a conjecture for trigonometric sums and starlike functions. J. Approx. Theory 149(1), 42–58 (2007)
Küstner, R.: On the order of starlikeness of the shifted Gauss hypergeometric function. J. Math. Anal. Appl. 334(2), 1363–1385 (2007)
Liu, J.-L.: On subordinations for certain multivalent analytic functions associated with the generalized hypergeometric function. J. Inequal. Pure Appl. Math. 7(4), 1–6 (2006)
Mondal, S.R., Swaminathan, A.: Coefficient conditions for univalency and starlikeness of analytic functions. J. Math. Appl. 31, 77–90 (2009)
Mondal, S.R., Swaminathan, A.: On the positivity of certain trigonometric sums and their applications. Comput. Math. Appl. 62(10), 3871–3883 (2011)
Mondal, S.R., Swaminathan, A.: Stable functions and extension of Vietoris’ theorem. Results Math. 62(1–2), 33–51 (2012)
Pommerenke, C.: Univalent Functions. Vandenhoeck and Ruprecht, Göttingen (1975)
Robertson, M.I.S.: On the theory of univalent functions. Ann. Math. (2) 37(2), 374–408 (1936)
Ruscheweyh, S.: Coefficient conditions for starlike functions. Glasg. Math. J. 29(1), 141–142 (1987)
Vietoris, L.: ber das Vorzeichen gewisser trignometrishcher Summen, Sitzungsber. Oest. Akad. Wiss. 167, 125–135 (1958)
Acknowledgments
The authors wish to thank the anonymous referees for their constructive criticism that helped improving this paper substantially. The first author is thankful to the “Council of Scientific and Industrial Research, India” (Grant Code: 09/143(0827)/2013-EMR-1) for financial support to carry out the above research work.
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by V. Ravichandran.
Rights and permissions
About this article
Cite this article
Sangal, P., Swaminathan, A. Starlikeness of Gaussian Hypergeometric Functions Using Positivity Techniques. Bull. Malays. Math. Sci. Soc. 41, 507–521 (2018). https://doi.org/10.1007/s40840-016-0420-5
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40840-016-0420-5
Keywords
- Trigonometric sums
- Starlike function
- Hypergeometric function
- Close-to-convex function
- Harmonic function
- Minimum principle