You may be suffering from a bacterial or yeast infection under that arm. Try putting some plain, full fat yogurt (make sure there is no sugar, fruit, or starch of any sort, and it should also contain live or active bacterial cultures) on the spot twice a day. Not only is it really good for your skin, but it will help restore the natural pH, and the friendly bacteria will reduce populations of harmful bacteria and yeasts.
Try using a deodorant that contains as few artificial ingredients as possible. I prefer Toms of Maine unscented, which contains coriander (which has been found to significantly reduce the numbers of pathogenic bacteria in certain studies).