According to certified personal trainer Robin Reichert for AZCentral, there are exercises specifically made to tone triceps, which in turn strengthens flabby underarms. These exercises should be performed using a weight machine. There is no best exercise, but the one that fits the exerciser's comfort level best.
Reichert states that 15 to 20 reps of the triceps pushdown exercise is an effective exercise for flabby underarms. Begin the exercise by keeping the elbows close to the body while grasping the hand attachments with an overhand grip. While squeezing the abdominal muscles and keeping the back straight, push downward until the arms are straight in one smooth action. Hold the stretched position for two to five seconds before returning the arms back to the starting position.
Reichert notes that the next exercise useful for remedying flabby underarms is the machine bench that utilizes weight or pulley machines that have resistance bands or bars. Face away from the pulleys or weights, sitting up in a straight position before grabbing the hand grips in a position so that the palms of each hand face each other. Push the hands away from the body until the arms are fully extended with the abdominal muscles squeezed. This exercise should be done 12 to 16 times.