ON SOME FAMILIES OF ANALYTIC FUNCTIONS

THEOREM 1. Any analytic function f in a Riemann surface R belongs to the Smirnov class S(R) if and only if the subharmonic function log +|/| has a quasi- bounded harmonic majorant in R. Using a version of Garding and Hόrmander's theorem [7] as a lemma, we shall prove: T H E O R E M 2. Any analytic function f in a Riemann surface R belongs to the Hardy class HP{R) (for p > 0 ) if and only if the subharmonic function \f\ p has a quasi-bounded harmonic majorant in R. Seeing the above characterizati

THEOREM 3. Let Ψ(r) be a continuous extended real-valued function defined for r > 0 satisfying the condition that for any finite positive real number c, the set of r such that the inequality Ψ(r) ^ c holds is bounded {from above). Let R be a Rie- mann surface, E be a closed polar set lying in R and f be an anaylytic function defined in R — E such that the composite function Ψ{\f\) has a quasi-bounded harmonic majorant in R — E. Then there exists an analytic function f defined in R such that the

LEMMA 1. Let υ be a quasi-bounded subharmonic function in a Riemann surface R. Then v is U.A.I, for arbitrary point z0 in R and arbitrary exhaustion {.Rv}> Zo in R\ Conversely assume that a subharmonic function v in R is U.A.I. for at least one point z0 and at least one exhaustion -[Rny > z^ in Rγ. Then v is a quasi-bounded subharmonic function in R.

LEMMA 2. A subharmonic function v is quasi-bounded if and only if there exists a non-negative monotone non-decreasing convex function Φ{r) defined for r >: 0 satisfying the conditions (i) and (ii).

Lemma 1. 3. Here we remark the relations between some families of analytic functions defined in a Riemann surface R. We define the families ΛB(R) and AL(R) of analytic functions in R by the following: / is in AB{R) if and only if \f\ is bounded in R; f is in AL(R) if and only if the subharmonic function log +|/| has a harmonic majorant in R. Then the following inclusion relations: AB{R) c HP[R) c S(R) c AL{R) (for p > 0)

COROLLARY 1. {An extended form of Theorem 1 in [17]) Any analytic function f is in the Smirnov class S{R) if and only if the subharmonic function log +|/| is U.A.I, for arbitrary fixed point z0 in R and arbitrary exhaustion {Rn}, z0 in Rx. https://doi.org/10.1017/S0027763000012630 Published online by Cambridge University Press

COROLLARY 2. {An extended form of Theorem 2 in [17]) Any analytic function f is in the Smirnov class S{R) if and only if the subharmonic function log +|/| has a harmonic majorant which is U.A.I, for arbitrary fixed point zQ in R and arbitrary exhaustion {Rn}, z0 in Rx. The following corollary shows that Gehrίng's class JV* in [8] is a special case of the Smirnov class S(R) where R is the unit open disc.

COROLLARY 3. Any analytic function f is in the class S(R) if and only if there exists a non-negative monotone non-decreasing convex function Φ{r) satisfying the condition (i) in §2 and the subharmonic function Φ(log + \f\) has a harmonic majorant in R.

LEMMA 3. Let v be a subharmonic function defined in R. Let φ{r) be a non-negative monotone non-decreasing convex function defined for — oo < r < + co satisfying the condition (A) lim φ(r) / r — + oo r —> ~\- oo and assume that (B) the subharmonic function φ{v) has a harmonic majorant in R, where we set φ(— oo) = lim φ(r). r -> — oo

LEMMA 4. Let E be a closed polar set in a Riemann surface R and assume that u is a quasi-bounded harmonic function defined in R — E. Then there exists a quasi-bounded harmonic function a defined in R such that the restriction of a to R — E coincides with u .

COROLLARY 1. {An extension of Tumarkin-HavinsorC s theorem [17]) Let E be a closed polar set lying in a Riemann surface R. If a function f is in the Smirnov class S(R — E), then there exists an analytic function f in the Smirnov class S(R) such that the restriction of f to R — E coincides with f.

COROLLARY 2. (Parreau [13], Theorem 20) Let E be a closed polar set lying in a Riemann surface R. If a function f is in the class HP{R — E) for p > 0, then there exists f in the class HP[R) such that the restriction of f to R — E coincides with f.