Abstract
In Ozaki (Sci Rep Tokyo Bunrika Daigaku Sect A 4:45–87, 1941) proved that if \(f(z)=z+a_2z^2+a_3z^3+\cdots\) is analytic in \(\mathbb {D}\), \(f'(z)\ne 0\) on \(|z|=r<1\), then the total variation of \(\arg \{f(z)\}\) on \(|z|=r\) is not greater than the total variation of \(\arg \{\mathrm{d}f(z)\}\) on \(|z|=r\), namely
We prove that without the integrals the above inequality implies \(f(z)=z\). Furthermore, we prove that if \(f'(z)\prec ((1+z)/(1-z))^2\), then f(z) is starlike in \(|z|<0.24\ldots\).
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1 Introduction
Let \(\mathcal {H}\) denote the class of functions analytic in the unit disk \(\mathbb {D}=\{z\in \mathbb {C}: |z|<1\}\). Let \(\mathcal {A}\) denote the class of functions \(f(z)\in \mathcal {H}\) of the form:
A set E is said to be convex if and only if it is starlike with respect to each of its points, that is if and only if the linear segment joining any two points of E lies entirely in E. Let \(f(z)\in \mathcal {H}\) and let f(z) be univalent in \(\mathbb {D}\). Then f(z) maps \(\mathbb {D}\) onto a convex domain if and only if
Such function f is said to be convex in \(\mathbb {D}\) (or briefly convex). Let \(\mathcal {K}\) denote the subclass of \({\mathcal {H}}\) consisting of functions satisfying 1.2 and normalized by \(f(0)=0\), \(f'(0)=1\). We denote by \({\mathcal {S}}\) the subclass of \({\mathcal {A}}\) consisting of univalent functions. A function \(f \in {\mathcal {S}}\) is said to be starlike of order \(\alpha\) if
for some \(0\le \alpha <1\). Condition (1.3) may be written as
where \(\prec\) denotes the subordination. If f(z) and g(z) are analytic in \(\mathbb {D}\), then we say that f(z) is subordinate to g(z) if there exists a function w(z), which is analytic in \(\mathbb {D}\) with \(w\left( 0\right) =0\) and \(\left| w\left( z\right) \right| <1\) such that
Therefore, \(f(z)\prec g(z)\) implies \(f(\mathbb {D})\subset g(\mathbb {D})\). Moreover, if the function g(z) is univalent in \(\mathbb {D}\), then
We denote by \({\mathcal {S}}^{*}(\alpha )\) the class of functions starlike of order \(\alpha\). An univalent function \(f\in \mathcal {S}\) belongs to the class of close-to-convex functions \(\mathcal {C}\) if and only if the complement E of the image-region \(F=\left\{ f(z): |z|<1\right\}\) is the union of rays that are disjoint (except that the origin of one ray may lie on another one of the rays). Equivalently, if \(f\in \mathcal {A}\) satisfies
for some \(g(z)\in \mathcal {S}^*\) and some \(\alpha \in (-\pi /2,\pi /2)\), then f(z) is close-to-convex (with respect to g(z)) in \(\mathbb {D}\) and denoted by \(f(z)\in \mathcal {C}\). It is known that \(\mathcal {C}\subset \mathcal {S}\) and it is known that if \(f(z)\in \mathcal {A}\) and f(z) satisfies (1.3), then f(z) is univalent in \(\mathbb {D}\).
A function f(z) which is analytic in a domain \(D \in \mathbb {C}\) is called p-valent in D if for every complex number w, the equation \(f(z)= w\) have at most p roots in D and there will be a complex number \(w_0\) such that the equation \(f(z) = w_0\), has exactly p roots in D. In Ozaki (1935) proved that if f(z) of the form \(f(z)=z^p+\sum _{n=p+1}^\infty a_nz^n\) and is analytic in a convex domain \(D\subset \mathbb {C}\) and for some real \(\alpha\) we have
then f(z) is at most p-valent in D. Ozaki’s condition is a generalization of the well-known Noshiro–Warschawski univalence condition with \(p=1\) (Noshiro 1934; Warschawski 1935). Let \(\mathcal {A}(p)\) be the class of functions of the form
where \(a_p\ne 0\) and \(p\in \mathbb {N}=\{1,2,\ldots \}\).
2 Main result
If
then \(f(z)=z+a_{n+1}z^{n+1}+\cdots\) is close to convex, (Avhadiev and Aksent’ev 1973). Umezawa in Umezawa (1955) proved that
implies the univalence of f(z) in \(|z|\le 1\). Notice also here that in Ozaki (1941) proved that if \(f(z)=z+a_2z^2+a_3z^3+\cdots\) is analytic in \(\mathbb {D}\), with \(f(z)f'(z)/z\ne 0\) there, and if either
or
holds throughout \(\mathbb {D}\), then f is univalent and convex in at least one direction in \(\mathbb {D}\). It has been generalized in Ogawa (1961), Shah (1973). The number \(\sqrt{6}\) in (2.1), was improved to \(3.05\ldots\) in Kudryashov (1973). Notice that the condition
\(0\le \alpha <2.832\ldots\) is sufficient for starlikeness, (Miller and Mocanu 2000, p.273). It is known that for \(f(z)\in \mathcal {A}\)
In Ozaki (1941) proved that if \(f(z)\in \mathcal {H}\), \(f'(z)\ne 0\) on \(|z|=r<1\), then the total variation of \(\arg \{f(z)\}\) on \(|z|=r\) is not greater than the total variation of \(\arg \{\mathrm{d}f(z)\}\) on \(|z|=r\), namely
or by a modification in the above inequality,
where \(z=re^{i\theta }\) and \(0\le \theta \le 2\pi\). The first our theorem describes a further relation between
For some new conditions for starlikeness and strongly starlikeness of order alpha, we refer to our recent paper (Nunokawa and Sokół (2017)).
Theorem 2.1
Let\(f(z)\in \mathcal {H}\), \(f(z)=z+\sum _{n=2}^\infty a_nz^n\), \(f'(z)\ne 0\)in\(\mathbb {D}\), and suppose that there exists\(r_0\), \(0<r_0\le 1\), such that
Then\(f(z)=z\)in\(\mathbb {D}\).
Proof
We have
so there exists a \(r_1,r_2\), \(r_1,r_2\in (0,1)\), such that
and
From the hypothesis of Theorem 2.1, we have
and therefore, for all r, \(0<r< R\)
where \(z=\rho e^{i\theta }\), and so
Then, from Ozaki’s theorem (2.4), we get
Therefore, we have
or
An interpretation above equality provides
where c is a real constant. Putting \(z=0\) gives us \(c=1\) or
Applying Cauchy–Riemann’s differential equations, we easily obtain
so, applying the Theorem of identity of analytic functions, we have \(f(z)=z\) for \(z\in \mathbb {D}\). It completes the proof. \(\square\)
Theorem 2.2
Let\(f(z)\in \mathcal {A}\), \(f'(z)\ne 0\)in\(\mathbb {D}\), and suppose that
Thenf(z) is starlike in\(|z|<r_0\), where\(r_0=0.246964\ldots\)is the smallest positive root of the equation
and\(\beta =0.38344486\ldots\)satisfies the condition
Proof
Subordination (2.7) follows that
In Nunokawa (1993), Nunokawa and Sokół (2017) it was proved that if \(p(z)\in \mathcal {H}\), \(p(0)=1\), \(p(z)\ne 0\) in \(\mathbb {D}\) and if there exists a point \(z_0\), \(|z_0|<1\), such that
and
for some \(\beta \in (0,2]\), then we have
where \(k\ge 1\), when \(\arg \left\{ p(z_0)\right\} =\pi \beta /2\) and \(k\le -1\), when \(\arg \left\{ p(z_0)\right\} =-\pi \beta /2\).
To apply this result for the proof of Theorem 2.2 assume that \(p(z)=f(z)/z\). For the value \(\beta =0.38344486\ldots\) if there exists a point \(z_0\), \(|z_0|=r_0<1\), such that
and
Then from (2.11), for the case \(\arg \{p(z_0)\}=\pi \beta /2\), we have
where \(k\ge 1\). Because
we have
This contradicts condition (2.10). For the case \(\arg \{p(z_0)\}=-\pi \beta /2\), applying the same method as above we can get
This is also contradicts condition (2.10). Therefore, we have
Applying the above results, we have
Putting
or
and
then we have
From (2.12) and (2.13), we can get \(\beta =0.38344486\ldots\) and \(r_0=0.246964\ldots\).
It completes the proof of Theorem 2.2. \(\square\)
Notice that if we take \(p(z)=f'(z)\) in the Theorem 2.2, then we obtain the following corollary.
Corollary 2.3
Let\(p(z)\in \mathcal {H}\), \(p(z)=1+p_1z+\cdots\), \(p(z)\ne 0\)in\(\mathbb {D}\), and suppose that
then
where\(r_0=0.246964\ldots\)is described in the hypothesis of Theorem2.2.
Theorem 2.4
Let\(f(z)\in \mathcal {A}\), \((zf'(z))'\ne 0\)in\(\mathbb {D}\), and suppose that
Then
where\(r_0=0.246964\ldots\)is described in the hypothesis of Theorem2.2. Condition (2.17) means thatf(z) is convex in\(|z|<r_0\).
Proof
The proof becomes trivial if we put \(p(z)=(zf'(z))'\) in Corollary 2.3. \(\square\)
Theorem 2.5
Let\(f(z)\in \mathcal {H}\), \(f(z)=z+\sum _{n=2}^\infty a_nz^n\), and suppose that there exists a starlike functiong(z) of order\(\alpha\)for which
Thenf(z) is starlike of order at least\(\alpha\) in \(\mathbb {D}\).
Proof
The above theorem is trivial, because (2.17) follows that
and so
This is exactly (1.4), so \(f(z)\in {\mathcal {S}}^{*}(\alpha )\). \(\square\)
References
Avhadiev FG, Aksent’ev LA (1973) The subordination principle in sufficient conditions for univalence. Dokl Akad Nauk SSSR 211(1):934–939 (Soviet Math. Dokl.14(4)(1973))
Kudryashov SN (1973) On some criteria of schlichtness of analytic functions (Russian). Mat. Zmetki 13:359–366
Miller SS, Mocanu PT (2000) Differential subordinations, theory and applications, series of monographs and textbooks in pure and applied mathematics, vol 225. Marcel Dekker Inc., New York
Noshiro K (1934–1935) On the theory of Schlicht functions. J Fac Sci Hokkaido Univ Jpn 2(1):129–135
Nunokawa M, Sokół J (2017) New conditions for starlikeness and strongly starlikeness of order alpha. Houston J Math 43(2):35–43
Nunokawa M (1993) On the order of strongly starlikeness of strongly convex functions. Proc Jpn Acad Ser A 69:234–237
Ogawa S (1961) Some criteria for univalence. J Nara Gakugei Univ No.1 10:7–12
Ozaki S (1935) On the theory of multivalent functions. Sci Rep Tokyo Bunrika Daigaku Sect A 2:167–188
Ozaki S (1941) On the theory of multivalent functions II. Sci Rep Tokyo Bunrika Daigaku Sect A 4:45–87
Shah GM (1973) On holomorphic functions convex in one direction. J Indian Math Soc 37:257–276
Umezawa T (1955) On the theory of univalent functions. Tohoku Math J 7:212–228
Warschawski S (1935) On the higher derivatives at the boundary in conformal mapping. Trans Am Math Soc 38:310–340
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Nunokawa, M., Sokół, J. Some Inequalities for Analytic Functions in the Unit Disc. Iran J Sci Technol Trans Sci 44, 773–777 (2020). https://doi.org/10.1007/s40995-020-00869-5
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DOI: https://doi.org/10.1007/s40995-020-00869-5